Permutation Patterns

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London Mathematical Society Lecture Note Series: 376

Permutation PatternsSt Andrews 2007

Edited bySTEVE LINTON

University of St Andrews

NIK RU S KUCUniversity of St Andrews

VINCENT VATTERDartmouth College

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore,So Paulo, Delhi, Dubai, Tokyo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-72834-8

ISBN-13 978-0-511-90182-9

Cambridge University Press 2010

2010

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Contents

Preface page 1

Some general results in combinatorial enumeration 3Martin Klazar

A survey of simple permutations 41Robert Brignall

Permuting machines and permutation patterns 67Mike Atkinson

On three dierent notions of monotone subsequences 89Miklos Bona

A survey on partially ordered patterns 115Sergey Kitaev

Generalized permutation patterns a short survey 137Einar Steingrmsson

An introduction to structural methods inpermutation patterns 153Michael Albert

Combinatorial properties of permutation tableaux 171Alexander Burstein and Niklas Eriksen

Enumeration schemes for words avoiding permutations 193Lara Pudwell

The lexicographic rst occurrence of a I-II-III-pattern 213Torey Burton, Anant P. Godbole and Brett M. Kindle

Enumeration of partitions by rises, levels and descents 221Touk Mansour and Augustine O. Munagi

Restricted patience sorting and barred pattern avoidance 233Alexander Burstein and Isaiah Lankham

Permutations with k-regular descent patterns 259Anthony Mendes, Jerey B. Remmel and Amanda Riehl

Packing rates of measures and a conjecture for the packingdensity of 2413 287Cathleen Battiste Presutti and Walter Stromquist

On the permutational power of token passing networks 317Michael Albert, Steve Linton and Nik Ruskuc

Problems and conjectures 339

Preface

The Permutation Patterns 2007 conference was held 1115 June 2007at the University of St Andrews. This was the fth Permutation Pat-terns conference; the previous conferences were held at Otago University(Dundein, New Zealand), Malaspina College (Vancouver Island, BritishColumbia), the University of Florida (Gainesville, Florida), and Reyk-javk University (Reykjavk, Iceland). The organizing committee wascomprised of Miklos Bona, Lynn Hynd, Steve Linton, Nik Ruskuc, EinarSteingrmsson, Vincent Vatter, and Julian West. A half-day excursionwas taken to Falls of Bruar, Blair Athol Castle and The Queens Viewon Loch Tummel.

There were two invited talks:

Mike Atkinson (Otago University, Dunedin, New Zealand), Simplepermutations and wreath-closed pattern classes.

Martin Klazar (Charles University, Prague, Czech Republic), Poly-nomial counting.

There were 35 participants, 23 talks, and a problem session (the prob-lems from which are included at the end of these proceedings). All themain strands of research in permutation patterns were represented, andwe hope this is reected by the articles of these proceedings, especiallythe eight surveys at the beginning. The conference was supported bythe EPSRC and Edinburgh Mathematical Society.

1

Some general results in combinatorialenumeration

Martin KlazarDepartment of Applied Mathematics

Charles University

118 00 Praha Czech Republic

Abstract

This survey article is devoted to general results in combinatorial enumer-ation. The rst part surveys results on growth of hereditary properties ofcombinatorial structures. These include permutations, ordered and un-ordered graphs and hypergraphs, relational structures, and others. Thesecond part advertises four topics in general enumeration: 1. countinglattice points in lattice polytopes, 2. growth of context-free languages, 3.holonomicity (i.e., P -recursiveness) of numbers of labeled regular graphsand 4. ultimate modular periodicity of numbers of MSOL-denablestructures.

1 Introduction

We survey some general results in combinatorial enumeration. A problemin enumeration is (associated with) an innite sequence P = (S1 , S2 , . . . )of nite sets Si . Its counting function fP is given by fP (n) = |Sn |, thecardinality of the set Sn . We are interested in results of the followingkind on general classes of problems and their counting functions.

Scheme of general results in combinatorial enumeration. Thecounting function fP of every problem P in the class C belongs to theclass of functions F . Formally, {fP | P C} F .

The larger C is, and the more specic the functions in F are, the strongerthe result. The present overview is a collection of many examples of thisscheme.

3

4 Klazar

One can distinguish general results of two types. In exact results,F is a class of explicitly dened functions, for example polynomials orfunctions dened by recurrence relations of certain type or functionscomputable in polynomial time. In asymptotic results, F consists offunctions dened by asymptotic equivalences or asymptotic inequali-ties, for example functions growing at most exponentially or functionsasymptotic to n(11/k)n+o(n) as n , with the constant k 2 beingan integer.

The sets Sn in P usually constitute sections of a xed innite set.Generally speaking, we take an innite universe U of combinatorialstructures and introduce problems and classes of problems as subsetsof U and families of subsets of U , by means of size functions s : U N0 = {0, 1, 2, . . . } and/or (mostly binary) relations between structuresin U . More specically, we will mention many results falling within theframework of growth of downsets in partially order sets, or posets.Downsets in posets of combinatorial structures. We consider

a nonstrict partial ordering (U,), where is a containment or a sub-structure relation on a set U of combinatorial structures, and a sizefunction s : U N0 . Problems P are downsets in (U,), meaning thatP U and A B P implies A P , and the counting function of Pis

fP (n) = #{A P | s(A) = n}.(More formally, the problem is the sequence of sections (P U1 , P U2 , . . . ) where Un = {A U | s(A) = n}.) Downsets are exactly thesets of the form

Av(F ) := {A U | A B for every B in F}, F U.There is a one-to-one correspondence P F = min(U\P ) and F P = Av(F ) between the family of downsets P and the family of an-tichains F , which are sets of mutually incomparable structures under .We call the antichain F = min(U\P ) corresponding to a downset P thebase of P .

We illustrate the scheme by three examples, all for downsets in posets.

1.1 Three examples

Example 1. Downsets of partitions. Here U is the family of par-titions of [n] = {1, 2, . . . , n} for n ranging in N, so U consists of nitesets S = {B1 , B2 , . . . , Bk} of disjoint and nonempty nite subsets Bi of

Some general results in combinatorial enumeration 5

N, called blocks, whose union B1 B2 Bk = [n] for some n inN. Two natural size functions on U are order and size, where the order,S, of S is the cardinality, n, of the underlying set and the size, |S|, ofS is the number, k, of blocks. The formula for the number of partitionsof [n] with k blocks

S(n, k) := #{S U | S = n, |S| = k} =k

i=0

(1)i(k i)ni!(k i)!

is a classical result (see [111]); S(n, k) are called Stirling numbers. It isalready a simple example of the above scheme but we shall go further.

For xed k, the function S(n, k) is a linear combination with rationalcoecients of the exponentials 1n , 2n , . . . , kn . So is the sum S(n, 1) +S(n, 2)+ +S(n, k) counting partitions with order n and size at most k.We denote the set of such partitions {S U | |S| k} as Uk . Considerthe poset (U,) with S T meaning that there is an increasing injectionf :

S T such that every two elements x, y in S lie in the same

block of S if and only if f(x), f(y) lie in the same block of T . In otherwords, S T means that T has a subset X of size S such thatT induces on X a partition order-isomorphic to S. Note that Uk isa downset in (U,). We know that the counting function of Uk withrespect to order n equals a11n + + akkn with ai in Q. What arethe counting functions of other downsets? If the size is bounded, asfor Uk , they have similar form as shown in the next theorem, provedby Klazar [77]. It is our rst example of an exact general enumerativeresult.

Theorem 1.1 (Klazar). If P is a downset in the poset of partitionssuch that maxSP |S| = k, then there exist a natural number n0 andpolynomials p1(x), p2(x), . . . , pk (x) with rational coecients such thatfor every n > n0 ,

fP (n) = #{S P | S = n} = p1(n)1n + p2(n)2n + + pk (n)kn .

If maxSP |S| = +, the situation is much more intricate and we are farfrom having a complete description but the growths of fP (n) below 2n1

have been determined (see Theorem 2.17 and the following comments).We briey mention three subexamples of downsets with unbounded size,none of which has fP (n) in the form of Theorem 1.1. If P consists of allpartitions of [n] into intervals of length at most 2, then fP (n) = Fn , thenth Fibonacci number, and so fP (n) = b1n + b2n where =

512 ,

6 Klazar

=5+12 and b1 =

5, b2 = 5 . If P is given as P = Av({C})

where C = {{1, 3}, {2, 4}} (the partitions in P are so called noncrossingpartition, see the survey of Simion [106]) then fP (n) = 1n+1

(2nn

), the nth

Catalan number which is asymptotically cn3/24n . Finally, if P = U ,so P consists of all partitions, then fP (n) = Bn , the nth Bell numberwhich grows superexponentially.Example 2. Hereditary graph properties. Here U is the uni-

verse of nite simple graphs G = ([n], E) with vertex sets [n], n rangingover N, and is the induced subgraph relation; G1 = ([n1 ], E1) G2 =([n2 ], E2) means that there is an injection from [n1 ] to [n2 ] (not necessar-ily increasing) that sends edges to edges and nonedges to nonedges. Thesize, |G|, of a graph G is the number of vertices. Problems are downsetsin (U,) and are called hereditary graph properties. The next theo-rem, proved by Balogh, Bollobas and Weinreich [18], describes countingfunctions of hereditary graph properties that grow no faster than expo-nentially.

Theorem 1.2 (Balogh, Bollobas and Weinreich). If P is a hereditarygraph property such that for some constant c > 1, fP (n) = #{G P | |G| = n} < cn for every n in N, then there exists a natural numbers kand n0 and polynomials p1(x), p2(x), . . . , pk (x) with rational coecientssuch that for every n > n0 ,

fP (n) = p1(n)1n + p2(n)2n + + pk (n)kn .

The case of superexponential growth of fP (n) is discussed below in The-orem 2.11.

In both examples we have the same class of functions F , linear com-binations p1(n)1n + p2(n)2n + + pk (n)kn with pi Q[x]. It would benice to nd a common extension of Theorems 1.1 and 1.2. It would bealso of interest to determine if the two classes of functions realizable ascounting functions in both theorems coincide and how they dier fromQ[x, 2x , 3x , . . . ].Example 3. Downsets of words. Here U is the set of nite words

over a nite alphabet A, so U = {u = a1a2 . . . ak | ai A}. Thesize, |u|, of such a word is its length k. The subword relation (alsocalled the factor relation) u = a1a2 . . . ak v = b1b2 . . . bl means thatbi+1 = a1 , bi+2 = a2 , . . . , bi+k = ak for some i. We associate with aninnite word v = b1b2 . . . over A the set P = Pv of all its nite subwords,thus Pv = {br br+1 . . . bs | 1 r s}. Note that Pv is a downset in


(U,). The next theorem was proved by Morse and Hedlund [92], seealso Allouche and Shallit [7, Theorem 10.2.6].

Theorem 1.3 (Morse and Hedlund). Let P be the set of all nite sub-words of an innite word v over a nite alphabet A. Then fP (n) =#{u P | |u| = n} is either larger than n for every n in N or iseventually constant. In the latter case the word v is eventually periodic.

The case when P is a general downset in (U,), not necessarily com-ing from an innite word (cf. Subsection 2.4), is discussed below inTheorem 2.19.

Examples 1 and 2 are exact results and example 3 combines a tightform of an asymptotic inequality with an exact result. Examples 1and 2 involve only countably many counting functions fP (n) and, asfollows from the proofs, even only countably many downsets P . In ex-ample 3 we have uncountably many distinct counting functions. Tosee this, take A = {0, 1} and consider innite words v of the formv = 10n1 10n2 10n3 1 . . . where 1 n1 < n2 < n2 < . . . is a sequenceof integers and 0m = 00 . . . 0 with m zeros. It follows that for distinctwords v the counting functions fPv are distinct; Proposition 2.1 presentssimilar arguments in more general settings.

1.2 Content of the overview

The previous three examples illuminated to some extent general enumer-ative results we are interested in but they are not fully representativebecause we shall cover a larger area than the growth of downsets. Wedo not attempt to set forth any more formalized denition of a generalenumerative result than the initial scheme but in Subsections 2.4 and3.4 we will discuss some general approaches of nite model theory basedon relational structures. Not every result or problem mentioned herets naturally the scheme; Proposition 2.1 and Theorem 2.6 are ratherresults to the eect that {fP | P C} is too big to be contained in asmall class F . This collection of general enumerative results is naturallylimited by the authors research area and his taste but we do hope thatit will be of interest to others and that it will inspire a quest for furthergeneralizations, strengthenings, renements, common links, unicationsetc.

For the lack of space, time and expertise we do not mention resultson growth in algebraic structures, especially the continent of growth ingroups; we refer the reader for information to de la Harpe [52] (and also

8 Klazar

to Cameron [43]). Also, this is not a survey on the class of problems#P in computational complexity theory (see Papadimitriou [94, Chap-ter 18]). There are other areas of general enumeration not mentionedproperly here, for example 0-1 laws (see Burris [42] and Spencer [109]).

In the next subsection we review some notions and denitions fromcombinatorial enumeration, in particular we recall the notion of Wilanformula (polynomial-time counting algorithm). In Section 2 we reviewresults on growth of downsets in posets of combinatorial structures. Sub-section 2.1 is devoted to pattern avoiding permutations, Subsections 2.2and 2.3 to graphs and related structures, and Subsection 2.4 to relationalstructures. Most of the results in Subsections 2.2 and 2.3 were found byBalogh and Bollobas and their coauthors [11, 13, 12, 15, 14, 16, 17, 18,19, 20, 21]. We recommend the comprehensive survey of Bollobas [30]on this topic. In Section 3 we advertise four topics in general enumer-ation together with some related results. 1. The EhrhartMacdonaldtheorem on numbers of lattice points in lattice polytopes. 2. Growthof context-free languages. 3. The theorem of Gessel on numbers of la-beled regular graphs. 4. The SpeckerBlatter theorem on numbers ofMSOL-denable structures.

1.3 Notation and some specic counting functions

As above, we write N for the set {1, 2, 3, . . . }, N0 for {0, 1, 2, . . . } and [n]for {1, 2, . . . , n}. We use #X and |X| to denote the cardinality of a set.By the phrase for every n we mean for every n in N and by for largen we mean for every n in N with possibly nitely many exceptions.Asymptotic relations are always based on n . The growth constantc = c(P ) of a problem P is c = lim sup fP (n)1/n ; the reciprocal 1/c isthen the radius of convergence of the power series

n0 fP (n)x

n .We review several counting sequences appearing in the mentioned re-

sults. Fibonacci numbers (Fn ) = (1, 2, 3, 5, 8, 13, . . . ) are given by therecurrence F0 = F1 = 1 and Fn = Fn1 + Fn2 for n 2. Theyare a particular case Fn = Fn,2 of the generalized Fibonacci numbersFn,k , given by the recurrence Fn,k = 0 for n < 0, F0,k = 1 andFn,k = Fn1,k + Fn2,k + + Fnk,k for n > 0. Using the nota-tion [xn ]G(x) for the coecient of xn in the power series expansion ofthe expression G(x), we have

Fn,k = [xn ]1

1 x x2 xk .


Standard methods provide asymptotic relations Fn,2 c2(1.618 . . . )n ,Fn,3 c3(1.839 . . . )n , Fn,4 c4(1.927 . . . )n and generally Fn,k cknkfor constants ck > 0 and 1 < k < 2; 1/k is the least positive root of thedenominator 1xx2 xk and 2 , 3 , . . . monotonically increaseto 2. The unlabeled exponential growth of tournaments (Theorem 2.21)is governed by the quasi-Fibonacci numbers F n dened by the recurrenceF 0 = F

1 = F

2 = 1 and F

n = F

n1 + F

n3 for n 3; so

F n = [xn ]

11 x x3

and F n c(1.466 . . . )n .We introduced Stirling numbers S(n, k) in Example 1. The Bell num-

bers Bn =n

k=1 S(n, k) count all partitions of an n-elements set andfollow the recurrence B0 = 1 and Bn =

n1k=0

(n1k

)Bk for n 1.

Equivalently,

Bn = [xn ]

k=0

xk

(1 x)(1 2x) . . . (1 kx) .

The asymptotic form of the Bell numbers is

Bn = nn(1log log n/ log n+O (1/ log n)) .

The numbers pn of integer partitions of n count the ways to expressn as a sum of possibly repeated summands from N, with the order ofsummands being irrelevant. Equivalently,

pn = [xn ]

k=1

11 xk .

The asymptotic form of pn is pn cn1 exp(dn) for some constants

c, d > 0. See Andrews [8] for more information on these asymptoticsand for recurrences satised by pn .

A sequence f : N C is a quasipolynomial if for every n we havef(n) = ak (n)nk + + a1(n)n + a0(n) where ai : N C are periodicfunctions. Equivalently,

f(n) = [xn ]p(x)

(1 x)(1 x2) . . . (1 xl)for some l in N and a polynomial p C[x]. We say that the sequence fis holonomic (other terms are P -recursive and D-nite) if it satises forevery n (equivalently, for large n) a recurrence

pk (n)f(n+ k) + pk1(n)f(n+ k 1) + + p0(n)f(n) = 0

10 Klazar

with polynomial coecients pi C[x], not all zero. Equivalently, thepower series

n0 f(n)x

n satises a linear dierential equation withpolynomial coecients. Holonomic sequences generalize sequences satis-fying linear recurrences with constant coecients. The sequences S(n, k),Fn,k , and F n for each xed k satisfy a linear recurrence with constant co-ecients and are holonomic. The sequences of Catalan numbers 1n+1

(2nn

)and of factorial numbers n! are holonomic as well. The sequences Bnand pn are not holonomic [112]. It is not hard to show that if (an ) isholonomic and every an is in Q, then the polynomials pi(x) in the re-currence can be taken with integer coecients. In particular, there areonly countably many holonomic rational sequences.

Recall that a power series F =

n0 anxn with an in C is alge-

braic if there exists a nonzero polynomial Q(x, y) in C[x, y] such thatQ(x, F (x)) = 0. F is rational if Q has degree 1 in y, that is, F (x) =R(x)/S(x) for two polynomials in C[x] where S(0) = 0. It is well known(Comtet [50], Stanley [112]) that algebraic power series have holonomiccoecients and that the coecients of rational power series satisfy (forlarge n) linear recurrence with constant coecients.Wilan formulas. A counting function fP (n) has a Wilan formula

(Wilf [117]) if there exists an algorithm that calculates fP (n) for everyinput n eectively, that is to say, in polynomial time. More precisely, werequire (extending the denition in [117]) that the algorithm calculatesfP (n) in the number of steps polynomial in the quantity

t = max(log n, log fP (n)).

This is (roughly) the minimum time needed for reading the input andwriting down the answer. In the most common situations when exp(nc) c > 0, this amountsto requiring a number of steps polynomial in n. But if fP (n) is small(say log n) or big (say doubly exponential in n), then one has to workwith t in place of n. The class of counting functions with Wilan formu-las includes holonomic sequences but is much more comprehensive thanthat.

2 Growth of downsets of combinatorial structures

We survey results in the already introduced setting of downsets in posetsof combinatorial structures (U,). The function fP (n) counts structuresof size n in the downset P and P can also be dened in terms of for-bidden substructures as P = Av(F ). Besides the containment relation


we employ also isomorphism equivalence relation on U and willcount unlabeled (i.e., nonisomorphic) structures in P . We denote thecorresponding counting function gP (n), so

gP (n) = #({A P | s(A) = n}/)

is the number of isomorphism classes of structures with size n in P .Restrictions on fP (n) and gP (n) dening the classes of functions F

often have the form of jumps in growth. A jump is a region of growthprohibited for counting functionsevery counting function resides ei-ther below it or above it. There are many kinds of jumps but the mostspectacular is perhaps the polynomialexponential jump from polyno-mial to exponential growth, which prohibits counting functions satisfy-ing nk < fP (n) < cn for large n for any constants k > 0 and c > 1. Forgroups, Grigorchuk constructed a nitely generated group having suchintermediate growth (Grigorchuk [70], Grigorchuk and Pak [69], [52]),which excludes the polynomialexponential jump for general nitely gen-erated groups, but a conjecture says that this jump occurs for everynitely presented group. We have seen this jump in Theorems 1.1 and1.2 (from polynomial growth to growth at least 2n ) and will meet newexamples in Theorems 2.4, 2.17, 2.18, 2.21, and 3.3.

If (U,) has an innite antichain A, then under natural conditionswe get uncountably many functions fP (n). This was observed severaltimes in the context of permutation containment and for completenesswe give the argument here again. These natural conditions, which willalways be satised in our examples, are niteness, for every n there arenitely many structures with size n in U , and monotonicity, s(G) s(H) & G H implies G = H for every G,H in U . (Recall that G Gfor every G.)

Proposition 2.1. If (U,) and the size function s() satisfy the mono-tonicity and niteness conditions and (U,) has an innite antichainA, then the set of counting functions fP (n) is uncountable.

Proof. By the assumption on U we can assume that the members ofA have distinct sizes. We show that all the counting functions fAv(F )for F A are distinct and so this set of functions is uncountable. Wewrite simply fF instead of fAv(F ) . If X,Y are two distinct subsets ofA, we express them as X = T {G} U and Y = T {H} V sothat, without loss of generality, m = s(G) < s(H), and G1 T,G2 Uimplies s(G1) < s(G) < s(G2) and similarly for Y (the sets T,U, V may

12 Klazar

be empty). Then, by the assumption on and s(),fX (m) = fT {G}(m) = fT (m) 1 = fT {H }V (m) 1 = fY (m) 1

and fX = fY .

An innite antichain thus gives not only uncountably many downsetsbut in fact uncountably many counting functions. Then, in particu-lar, almost all counting functions are not computable because we haveonly countably many algorithms. Recently, Albert and Linton [4] signif-icantly rened this argument by showing how certain innite antichainsof permutations produce even uncountably many growth constants, seeTheorem 2.6.

On the other hand, if every antichain is nite then there are onlycountably many functions fP (n). Posets with no innite antichain arecalled well quasiorderings or shortly wqo. (The second part of the wqoproperty, nonexistence of innite strictly descending chains, is satisedautomatically by the monotonicity condition.) But even if (U,) hasinnite antichains, there still may be only countably many downsetsP with slow growth functions fP (n). For example, this is the case inTheorems 1.1 and 1.2. It is then of interest to determine for whichgrowth uncountably many downsets appear (cf. Theorem 2.5). Theposets (U,) considered here usually have innite antichains, with twonotable wqo exceptions consisting of the minor ordering on graphs andthe subsequence ordering on words over a nite alphabet.

2.1 Permutations

Let U denote the universe of permutations represented by nite se-quences b1b2 . . . bn such that {b1 , b2 , . . . , bn} = [n]. The size of a permu-tation = a1a2 . . . am is its length || = m. The containment relation onU is dened by = a1a2 . . . am = b1b2 . . . bn if and only if for someincreasing injection f : [m] [n] one has ar < as bf (r) < bf (s) forevery r, s in [m]. Problems P are downsets in (U,) and their countingfunctions are fP (n) = #{ P | || = n}. The poset of permutations(U,) has innite antichains (see Aktinson, Murphy, and Ruskuc [10]).For further information and background on the enumeration of downsetsof permutations see Bona [34].

Recall that c(P ) = lim sup fP (n)1/n . We dene

E = {c(P ) [0,+] | P is a downset of permutations}


to be the set of growth constants of downsets of permutations. E con-tains elements 0, 1 and + because of the downsets , {(1, 2, . . . , n) | n N} and U (all permutations), respectively. How much does fP (n) dropfrom fU (n) = n! if P = U? The StanleyWilf conjecture (Bona [33, 34])asserted that it drops to exponential growth. The conjecture was provedin 2004 by Marcus and Tardos [87].

Theorem 2.2 (Marcus and Tardos). If P is a downset of permutationsthat is not equal to the set of all permutations, then, for some constantc, fP (n) < cn for every n.

Thus, with the sole exception of U , every P has a nite growth constant.Arratia [9] showed that if F consists of a single permutation then c(P ) =c(Av(F )) is attained as a limit lim fP (n)1/n . It would be nice to extendthis result.

Problem 2.3. Does lim fP (n)1/n always exist when F in P = Av(F )has more than one forbidden permutation?

For innite F there conceivably might be oscillations between two dier-ent exponential growths (similar oscillations occur for hereditary graphproperties and for downsets of words). It would be surprising if oscilla-tions occurred for nite F .

Kaiser and Klazar [76] determined growths of downsets of permuta-tions in the range up to 2n1 .

Theorem 2.4 (Kaiser and Klazar). If P is a downset of permutations,then exactly one of the following four cases occurs.

(i) For large n, fP (n) is constant.(ii) There are integers a0 , . . . , ak , k 1 and ak > 0, such that

fP (n) = a0(n0

)+ + ak

(nk

)for large n. Moreover, fP (n) n

for every n.(iii) There are constants c, k in N, k 2, such that Fn,k fP (n)

ncFn,k for every n, where Fn,k are the generalized Fibonacci num-bers.

(iv) One has fP (n) 2n1 for every n.The lower bounds in cases 2, 3, and 4 are best possible.

This implies that

E [0, 2] = {0, 1, 2, 2 , 3 , 4 , . . . },k being the growth constants of Fn,k , and that lim fP (n)1/n exists and

14 Klazar

equals to 0, 1 or to some k whenever fP (n) < 2n1 for one n. Notethat 2 is the single accumulation point of E [0, 2]. We shall see thatTheorem 2.4 is subsumed in Theorem 2.17 on ordered graphs. Case 2and case 3 with k = 2 give the polynomialFibonacci jump: If P is adownset of permutations, then either fP (n) grows at most polynomially(and in fact equals to a polynomial for large n) or at least Fibonac-cially. Huczynska and Vatter [72] gave a simpler proof for this jump.Theorem 2.18 extends it to edge-colored cliques. Theorem 2.4 combinesan exact result in case 1 and 2 with an asymptotic result in case 3. Itwould be nice to have an exact result in case 3 as well and to deter-mine precise forms of the corresponding functions fP (n) (it is knownthat in cases 13 the generating function

n0 fP (n)x

n is rational, seethe remarks at the end of this subsection). Klazar [80] proved thatcases 13 comprise only countably many downsets, more precisely: iffP (n) < 2n1 for one n, then P = Av(F ) has nite base F . In the otherdirection he showed [80] that there are uncountably many downsets Pwith fP (n) < (2.336 . . . )n for large n. Recently, Vatter [115] determinedthe uncountability threshold precisely and extended the description ofE above 2.

Theorem 2.5 (Vatter). Let = 2.205 . . . be the real root of x32x21.There are uncountably many downsets of permutations P with c(P ) but only countably many of them have c(P ) < and for each of theselim fP (n)1/n exists. Moreover, the countable intersection

E (2, )consists exactly of the largest positive roots of the polynomials

(i) 3 + 2x+ x2 + x3 x4 ,(ii) 1 + 3x+ 2x2 + x3 + x4 x5 ,(iii) 1 + 3x+ 2x2 + x3 + x4 x5 ,(iv) 1 + 3x+ x2 + x3 x4

and the two families (k, l range over N)

(v) 1 + xl xk+ l 2xk+ l+2 + xk+ l+3 , and(vi) 1 xk 2xk+2 + xk+3 .

The set E (2, ) has no accumulation point from above but it hasinnitely many accumulation points from below: is the smallest ele-ment of E which is an accumulation point of accumulation points. Thesmallest element of E (2, ) is 2.065 . . . (k = l = 1 in the family (v)).


In [12] it was conjectured that all elements of E (even in the moregeneral situation of ordered graphs) are algebraic numbers and that Ehas no accumulation point from above. These conjectures were refutedby Albert and Linton [4]. Recall that a subset of R is perfect if it isclosed and has no isolated point. Due to the completeness of R such aset is inevitably uncountable.

Theorem 2.6 (Albert and Linton). The set E of growth constants ofdownsets of permutations contains a perfect subset and therefore is un-countable. Also, E contains accumulation points from above.

The perfect subset constructed by Albert and Linton has smallest ele-ment 2.476 . . . and they conjectured that E contains some real interval(,+), which has been recently established by Vatter [114]. However,a typical downset produced by their construction has innite base. Itseems that the refuted conjectures should have been phrased for nitelybased downsets.

Problem 2.7. Let E be the countable subset of E consisting of thegrowth constants of nitely based downsets of permutations. Show thatevery in E is an algebraic number and that for every in E thereis a > 0 such that (, + ) E = .We know from [115] that E [0, ) = E [0, ).

We turn to the questions of exact counting. In view of Proposition 2.1and Theorem 2.6, we restrict to downsets of permutations with nitebases. The next problem goes back to Gessel [67, the nal section].

Problem 2.8. Is it true that for every nite set of permutations F thecounting function fAv(F )(n) is holonomic?

All explicit fP (n) found so far are holonomic. Zeilberger conjectures(see [57]) that P = Av(1324) has nonholonomic counting function (seeMarinov and Radoicic [88] and Albert et al. [3] for the approaches tocounting Av(1324)). We remarked earlier that almost all innitely basedP have nonholonomic fP (n).

More generally, one may pose (Vatter [116]) the following question.

Problem 2.9. Is it true that for every nite set of permutations Fthe counting function fAv(F )(n) has a Wilan formula, that is, can beevaluated by an algorithm in number of steps polynomial in n?

Wilan formulas were shown to exist for several classes of nitely baseddownsets of permutations. We refer the reader to Vatter [116] for further

16 Klazar

information and mention here only one such result due to Albert andAtkinson [1]. Recall that = a1a2 . . . an is a simple permutation if{ai, ai+1 , . . . , aj} is not an interval in [n] for every 1 i j n,0 < j i < n 1.

Theorem 2.10 (Albert and Atkinson). If P is a downset of permu-tations containing only nitely many simple permutations, then P isnitely based and the generating function

n0 fP (n)x

n is algebraicand thus fP (n) has a Wilan formula.

Brignall, Ruskuc and Vatter [41] show that it is decidable whether adownset given by its nite basis contains nitely many simple permuta-tions and Brignall, Huczynska and Vatter [40] extend Theorem 2.10 byshowing that many subsets of downsets with nitely many simple permu-tations have algebraic generating functions as well. See also Brignallssurvey [39].

We conclude this subsection by looking back at Theorems 2.4 and2.5 from the standpoint of eectivity. Let a downset of permutationsP = Av(F ) be given by its nite base F . Then it is decidable whetherc(P ) < 2 and (as noted in [115]) for c(P ) < 2 the results of Albert, Lintonand Ruskuc [5] provide eectively a Wilan formula for fP (n), in fact,the generating function is rational. Also, it is decidable whether fP (n)is a polynomial for large n (see [72], Albert, Atkinson and Brignall [2]).By [115], it is decidable whether c(P ) < and Vatter conjectures thateven for c(P ) < the generating function of P is rational.

2.2 Unordered graphs

U is the universe of nite simple graphs with normalized vertex sets[n] and is the induced subgraph relation. Problems P are heredi-tary graph properties, that is, downsets in (U,), and fP (n) counts thegraphs in P with n vertices. Monotone properties, which are hereditaryproperties that are closed under taking any subgraph, constitute a morerestricted family. An even more restricted family consists of minor-closedclasses, which are monotone properties that are closed under contract-ing edges. By Proposition 2.1 there are uncountably many countingfunctions of monotone properties (and hence of hereditary properties aswell) because, for example, the set of all cycles is an innite antichainunder the subgraph ordering. On the other hand, by the monumentaltheorem of Robertson and Seymour [100] there are no innite antichains


in the minor ordering and so there are only countably many minor-closed classes. The following remarkable theorem describes growths ofhereditary properties.

Theorem 2.11 (Balogh, Bollobas, Weinreich, Alekseev, Thomason). IfP is a proper hereditary graph property then exactly one of the four casesoccurs.

(i) There exist rational polynomials p1(x), p2(x), . . . , pk (x) such thatfP (n) = p1(n) + p2(n)2n + + pk (n)kn for large n.

(ii) There is a constant k in N, k 2, such that fP (n) = n(11/k)n+o(n)for every n.

(iii) One has nn+o(n) < fP (n) < 2o(n2 ) for every n.

(iv) There is a constant k in N, k 2, such that for every n we havefP (n) = 2(1/21/2k)n

2 +o(n2 ).

We mentioned case 1 as Theorem 1.2. The rst three cases were provedby Balogh, Bollobas and Weinreich in [18]. The fourth case is due toAlekseev [6] and independently Bollobas and Thomason [31].

Now we will discuss further strengthenings and renements of Theo-rem 2.11. Scheinerman and Zito in a pioneering work [103] obtained itsweaker version. They showed that for a hereditary graph property Peither (i) fP (n) is constantly 0, 1 or 2 for large n or (ii) ank < fP (n) ncn for large n for every constant c > 0.

In cases 1, 2, and 4 growths of fP (n) settle to specic asymptoticvalues and these can be characterized by certain minimal hereditaryproperties, as shown in [18]. Case 3, the penultimate rate of growth(see [19]), is very dierent. Balogh, Bollobas and Weinreich provedin [19] that for every c > 1 and > 1/c there is a monotone property Psuch that

fP (n) [ncn+o(n) , 2(1+o(1))n2 ]for every n and fP (n) attains either extremity of the interval innitelyoften. Thus in case 3 the growth may oscillate (innitely often) betweenthe bottom and top parts of the range. The paper [19] contains furtherexamples of oscillations (we stated here just one simplied version) anda conjecture that for nite F the functions fAv(F )(n) do not oscillate.As for the upper boundary of the range, in [19] it is proven that for every

18 Klazar

monotone property P ,

fP (n) = 2o(n2 ) fP (n) < 2n21 / t+ o ( 1 ) for some t in N.

For hereditary properties this jump is only conjectured. What about thelower boundary? The paper [21] is devoted to the proof of the followingtheorem.

Theorem 2.12 (Balogh, Bollobas and Weinreich). If P is a hereditarygraph property, then exactly one of the two cases occurs.

(i) There is a constant k in N such that fP (n) < n(11/k)n+o(n) forevery n.

(ii) For large n, one has fP (n) Bn where Bn are Bell numbers.This lower bound is the best possible.

By this theorem, the growth of Bell numbers is the lower boundary ofthe penultimate growth in case 3 of Theorem 2.11.

Monotone properties of graphs are hereditary and therefore their count-ing functions follow Theorem 2.11. Their more restricted nature allowsfor simpler proofs and simple characterizations of minimal monotoneproperties, which is done in the paper [20]. Certain growths of heredi-tary properties do not occur for monotone properties, for example if Pis monotone and fP (n) is unbounded, then fP (n)

(n2

)+ 1 for every

n (see [20]) but, P consisting of complete graphs with possibly an ad-ditional isolated vertex is a hereditary property with fP (n) = n+ 1 forn 3. More generally, Balogh, Bollobas and Weinreich show in [20]that if P is monotone and fP (n) grows polynomially, then

fP (n) = a0

(n

0

)+ a1

(n

1

)+ + ak

(n

k

)for large n

and some integer constants 0 aj 2j (j1)/2 . In fact, [20] deals mostlywith general results on the extremal functions eP (n) := max{|E| | G =([n], E) P} for monotone properties P .

For the top growths in case 4 of Theorem 2.11, Alekseev [6] and Bol-lobas and Thomason [32] proved that for P = Av(F ) with fP (n) =2(1/21/2k)n

2 +o(n2 ) the parameter k is equal to the maximum r suchthat there is an s, 0 s r, with the property that no graph in F canhave its vertex set partitioned into r (possibly empty) blocks inducing scomplete graphs and r s empty graphs. For monotone properties thisreduces to k = min{(G) 1 | G F} where is the usual chromaticnumber of graphs. Balogh, Bollobas and Simonovits [17] replaced for


monotone properties the error term o(n2) by O(n ), = (F ) < 2.Ishigami [74] recently extended case 4 to k-uniform hypergraphs.

Minor-closed classes of graphs were recently looked at from the pointof view of counting functions as well. They again follow Theorem 2.11,with possible simplications due to their more restricted nature. Oneis that there are only countably many minor-closed classes. Anothersimplication is that, with the trivial exception of the class of all graphs,case 4 does not occur as proved by Norine et al. [93].

Theorem 2.13 (Norine, Seymour, Thomas and Wollan). If P is aproper minor-closed class of graphs then fP (n) < cnn! for every n inN for a constant c > 1.

Bernardi, Noy and Welsh [26] obtained the following theorem; weshorten its statement by omitting characterizations of classes P withthe given growth rates.

Theorem 2.14 (Bernardi, Noy and Welsh). If P is a proper minor-closed class of graphs then exactly one of the six cases occurs.

(i) The counting function fP (n) is constantly 0 or 1 for large n.(ii) For large n, fP (n) = p(n) for a rational polynomial p of degree

at least 2.(iii) For every n, 2n1 fP (n) < cn for a constant c > 2.(iv) There exist constants k in N, k 2, and 0 < a < b such that

ann(11/k)n < fP (n) < bnn(11/k)n for every n.(v) For every n, Bn fP (n) = o(1)nn! where Bn is the nth Bell

number.(vi) For every n, n! fP (n) < cnn! for a constant c > 1.

The lower bounds in cases 3, 5 and 6 are best possible.

In fact, in case 3 the formulas of Theorem 1.2 apply. Using the stronglyrestricted nature of minor-closed classes, one could perhaps obtain incase 3 an even more specic exact result. The work in [26] gives furtherresults on the growth constants lim (fP (n)/n!)1/n in case 6 and statesseveral open problems, of which we mention the following analogue ofTheorem 2.2 for unlabeled graphs. A similar conjecture was also madeby McDiarmid, Steger and Welsh [89].

Problem 2.15. Does every proper minor-closed class of graphs containat most cn nonisomorphic graphs on n vertices, for a constant c > 1?

20 Klazar

This brings us to the unlabeled count of hereditary properties. Thefollowing theorem was obtained by Balogh et al. [16].

Theorem 2.16 (Balogh, Bollobas, Saks and Sos). If P is a heredi-tary graph property and gP (n) counts nonisomorphic graphs in P by thenumber of vertices, then exactly one of the three cases occurs.

(i) For large n, gP (n) is constantly 0, 1 or 2.(ii) For every n, gP (n) = cnk + O(nk1) for some constants k in N

and c in Q, c > 0.(iii) For large n, gP (n) pn where pn is the number of integer parti-

tions of n. This lower bound is best possible.

(We have shortened the statement by omitting the characterizations ofP with given growth rates.) The authors of [16] remark that with moreeort case 2 can be strengthened, for large n, to an exact result withthe error term O(nk1) replaced by a quasipolynomial p(n) of degree atmost k 1. It turns out that a weaker form of the jump from case 2 tocase 3 was proved already by Macpherson [86, 85]: If G = (N, E) is aninnite graph and gG (n) is the number of its unlabeled n-vertex inducedsubgraphs then either gG (n) nc for every n and a constant c > 0 orgG (n) > exp(n1/2) for large n for every constant > 0. Pouzet [96]showed that in the former case c1nd < gG (n) < c2nd for every n andsome constants 0 < c1 < c2 and d in N.

2.3 Ordered graphs and hypergraphs, edge-colored cliques,

words, posets, tournaments, and tuples

Ordered graphs. As previously, U is the universe of nite simplegraphs with vertex sets [n] but is now the ordered induced subgraphrelation, which means that G1 = ([m], E1) G2 = ([n], E2) if and onlyif there is an increasing injection f : [m] [n] such that {u, v} E1 {f(u), f(v)} E2 . Problems are downsets P in (U,), arecalled hereditary properties of ordered graphs, and fP (n) is the numberof graphs in P with vertex set [n]. The next theorem, proved by Balogh,Bollobas and Morris [12], vastly generalizes Theorem 2.4.

Theorem 2.17 (Balogh, Bollobas and Morris). If P is a hereditaryproperty of ordered graphs, then exactly one of the four cases occurs.

(i) For large n, fP (n) is constant.


(ii) There are integers a0 , . . . , ak , k 1 and ak > 0, such thatfP (n) = a0

(n0

)+ + ak

(nk

)for large n. Moreover, fP (n) n

for every n.(iii) There are constants c, k in N, k 2, such that Fn,k fP (n)

ncFn,k for every n, where Fn,k are the generalized Fibonacci num-bers.

(iv) One has fP (n) 2n1 for every n.The lower bounds in cases 2, 3, and 4 are best possible.

This is an extension of Theorem 2.4 because the poset of permutationsis embedded in the poset of ordered graphs via representing a permuta-tion = a1a2 . . . an by the graph G = ([n], {{i, j} | i < j & ai < aj}).One can check that G G and that the graphs Gform a downset in the poset of ordered graphs, so Theorem 2.17 impliesTheorem 2.4. Similarly, the poset of set partitions of Example 1 in In-troduction is embedded in the poset of ordered graphs, via representingpartitions by graphs whose components are cliques. Thus the growthsof downsets of set partitions in the range up to 2n1 are described byTheorem 2.17. As for permutations, it would be nice to have in case 3an exact result. Balogh, Bollobas and Morris conjecture that 2.031 . . .(the largest real root of x5x4x3x22x1) is the smallest growthconstant for ordered graphs above 2 and Vatter [115] notes that this isnot an element of E and thus here the growth constants for permutationsand for ordered graphs part ways.Edge-colored cliques. Klazar [81] considered the universe U of pairs

(n, ) where n ranges over N and is a mapping from the set([n ]

2

)of

two-element subsets of [n] to a nite set of colors C. The containment is dened by (m,) (n, ) if and only if there is an increasing injectionf : [m] [n] such that ({f(x), f(y)}) = ({x, y}) for every x, y in [m].For two colors we recover ordered graphs with induced ordered subgraphrelation. In [81] the following theorem was proved.

Theorem 2.18 (Klazar). If P is a downset of edge-colored cliques, thenexactly one of the three cases occurs.

(i) The function fP (n) is constant for large n.(ii) There is a constant c in N such that n fP (n) nc for every n.(iii) One has fP (n) Fn for every n, where Fn are the Fibonacci

numbers.

The lower bounds in cases 2 and 3 are best possible.

22 Klazar

This extends the bounded-linear jump and the polynomial-Fibonaccijump of Theorem 2.17. It would be interesting to have full Theo-rem 2.17 in this more general setting. As explained in [81], many posetsof structures can be embedded in the poset of edge-colored cliques (aswe have just seen for permutations) and thus Theorem 2.18 applies tothem. With more eort, case 2 can be strengthened to the exact resultfP (n) = p(n) with rational polynomial p(x).Words over nite alphabet. We revisit Example 3 from Introduc-

tion. Recall that U = A consists of all nite words over nite alphabetA and is the subword ordering. This ordering has innite antichains,for example 11, 101, 1001, . . . for A = {0, 1}. Balogh and Bollobas [11]investigated general downsets in (A,) and proved the following ex-tension of Theorem 1.3.

Theorem 2.19 (Balogh and Bollobas). If P is a downset of nite wordsover a nite alphabet A in the subword ordering, then fP (n) is eitherbounded or fP (n) n+ 1 for every n.In contrast with Theorem 1.3, for general downsets, a bounded functionfP (n) need not be eventually constant. Balogh and Bollobas [11] showedthat for xed s in N function fP (n) may oscillate innitely often betweenthe maximum and minimum values s2 and 2s 1, and s2 + s and 2s.These are, however, the wildest bounded oscillations possible since theyproved, as their main result, that if fP (n) = m n for some n thenfP (N) (m+1)2/4 for every N , N n+m. They also gave examplesof unbounded oscillations of fP (n) between n+ g(n) and 2n/g(n) for anyincreasing and unbounded function g(n) = o(log n), with the downset Pcoming from an innite word over two-letter alphabet.

Another natural ordering on A is the subsequence ordering wherea1a2 . . . ak b1b2 . . . bl if and only if bi1 = a1 , bi2 = a2 , . . . , bik = ak forsome indices 1 i1 < i2 < < ik l. Downsets in this orderingremain downsets in the subword ordering and thus their counting func-tions are governed by Theorem 2.19. But they can be also embeddedin the poset of edge-colored complete graphs (associate with a1a2 . . . anthe pair (n, ) where ({i, j}) = {ai, aj} for i < j) and Theorem 2.18applies. In particular, if P A is a downset in the subsequence or-dering, then fP (n) is constant for large n or fP (n) n + 1 for everyn (by Theorems 2.18 and 2.19). The subsequence ordering on A is awqo by Higmans theorem [71] and therefore has only countably manydownsets.

A variation on the subsequence ordering is the ordering on A given


by u = a1a2 . . . ak v = b1b2 . . . bl if and only if there is a permutation of the alphabet A such that a1a2 . . . ak (b1)(b2) . . . (bl) in thesubsequence ordering, that is, u becomes a subsequence of v after theletters in v are injectively renamed. This ordering on A gives Example1 in Introduction and leads to Theorem 1.1. It is a wqo as well.Posets and tournaments. U is the set of all pairs S = ([n],S )

where S is a non-strict partial ordering on [n]. We set R = ([m],R) S = ([n],S ) if and only if there is an injection f : [m] [n] suchthat x R y f(x) S f(y) for every x, y in [m]. Thus R Smeans that the poset R is an induced subposet of S. Downsets in (U,), hereditary properties of posets, and their growths were investigatedby Balogh, Bollobas and Morris in [14]. For the unlabeled count theyobtained the following result.

Theorem 2.20 (Balogh, Bollobas and Morris). If P is a hereditaryproperty of posets and gP (n) counts nonisomorphic posets in P by thenumber of vertices, then exactly one of the three cases occurs.

(i) Function gP (n) is bounded.(ii) There is a constant c > 0 such that, for every n,

n+12

gP (n)

n+12

+ c.

(iii) For every n, gP (n) n.

The lower bounds in cases 2 and 3 are best possible.

As for the labeled count fP (n), using case 1 of Theorem 2.11 they provedin [14, Theorem 2] that if P is a hereditary property of posets theneither (i) fP (n) is constantly 1 for large n or (ii) there are k integersa1 , . . . , ak , ak = 0, such that fP (n) = a0

(n0

)+ + ak

(nk

)for large n or

(iii) fP (n) 2n 1 for every n, n 6. Moreover, the lower bound incase (iii) is best possible and in case (ii) one has fP (n)

(n0

)+ + (nk)

for every n, n 2k + 1 and this bound is also best possible.A tournament is a pair T = ([n], T ) where T is a binary relation on [n]

such that xTx for no x in [n] and for every two distinct elements x, y in[n] exactly one of xTy and yTx holds. U consists of all tournaments forn ranging in N and is the induced subtournament relation. Balogh,Bollobas and Morris considered in [15, 14] unlabeled counting functionsof hereditary properties of tournaments. We merge their results in asingle theorem.

Theorem 2.21 (Balogh, Bollobas and Morris). If P is a hereditary

24 Klazar

property of tournaments and gP (n) counts nonisomorphic tournamentsin P by the number of vertices, then exactly one of the three cases occurs.

(i) For large n, function gP (n) is constant.(ii) There are constants k in N and 0 < c < d such that cnk cn for large n for a constant c > 1.Ordered hypergraphs. U consists of all hypergraphs, which are

the pairs H = ([n],H) with n in N and H being a set of nonempty andnon-singleton subsets of [n], called edges. Note that U extends both theuniverse of nite simple graphs and the universe of set partitions. Thecontainment is ordered but non-induced and is dened by ([m], G) ([n],H) if and only if there is an increasing injection f : [m] [n] and aninjection g : G H such that for every edge E in G we have f(E) g(E). Equivalently, one can omit some vertices from [n], some edgesfrom H and delete some vertices from the remaining edges in H so thatthe resulting hypergraph is order-isomorphic to G. Downsets in (U,)are called strongly monotone properties of ordered hypergraphs. Again,fP (n) counts hypergraphs in P with the vertex set [n]. We dene aspecial downset : we associate with every permutation = a1a2 . . . anthe (hyper)graph G = ([2n], {{i, n+ai} | i [n]}) and let denote theset of all hypergraphs in U contained in some graph G ; the graphs in dier from G s only in adding in all ways isolated vertices. Note that for two permutations if and only if G G for the corresponding(hyper)graphs. The next theorem was conjectured by Klazar in [77] forset partitions and in [78] for ordered hypergraphs.

Theorem 2.22 (Balogh, Bollobas, Morris, Klazar, Marcus). If P is astrongly monotone property of ordered hypergraphs, then exactly one ofthe two cases occurs.


(i) There is a constant c > 1 such that fP (n) cn for every n.(ii) One has P , which implies that

fP (n) n/2k=0

(n

2k

)k! = nn+O (n/ log n)

for every n and that the lower bound is best possible.

The theorem was proved by Balogh, Bollobas and Morris [13] and inde-pendently by Klazar and Marcus [81] (by means of results from [79, 78,87]). It follows that Theorem 2.22 implies Theorem 2.2. Theorem 2.22was motivated by eorts to extend the StanleyWilf conjecture, now theMarcusTardos theorem, from permutations to more general structures.A further extension would be to have it for the wider class of hereditaryproperties of ordered hypergraphs. These correspond to the containment dened by ([m], G) ([n],H) if and only if there is an increasing in-jection f : [m] [n] such that {f(E) | E G} = {f([m])E | E H}.The following conjecture was proposed in [13].

Problem 2.23. If P is a hereditary property of ordered hypergraphs,then either fP (n) cn for every n for some constant c > 1 or one hasfP (n)

n/2k=0

(n2k

)k! for every n. Moreover, the lower bound is best

possible.

As noted in [13], now it is no longer true that in the latter case P mustcontain .Tuples of nonnegative integers. For a xed k in N, we set U =

Nk0 , so U contains all k-tuples of nonnegative integers. We dene thecontainment by a = (a1 , a2 , . . . , ak ) b = (b1 , b2 , . . . , bk ) if and only ifai bi for every i. By Higmans theorem, (U,) is wqo. So there areonly countably many downsets. The size function on U is givenby a = a1 + a2 + + ak . For a downset P in (U,), fP (n) countsall tuples in P whose entries sum up to n. Stanley [110] (see also [111,Exercise 6 in Chapter 4]) obtained the following result.

Theorem 2.24 (Stanley). If P is a downset of k-tuples of nonnegativeintegers, then there is a rational polynomial p(x) such that fP (n) = p(n)for large n.

In fact, the theorem holds for upsets as well because they are comple-ments of downsets and #{a Nk0 | a = n} =

(n+k1k1

)is a rational

polynomial in n. Jelnek and Klazar [75] noted that the theorem holds

26 Klazar

for the larger class of sets P Nk0 that are nite unions of the general-ized orthants {a Nk0 | ai i bi , i [k]}, here (b1 , . . . , bk ) is in Nk0 andeach i is either or equality =; we call such P simple sets. It appearsthat this generalization of Theorem 2.24 provides a unied explanationof the exact polynomial results in Theorems 1.1, 2.4, 2.11, and 2.17, bymapping downsets of structures in a size-preserving manner onto simplesets in Nk0 for some k.

2.4 Growths of proles of relational structures

In this subsection we mostly follow the survey article of Pouzet [97], seealso Cameron [44]. This approach of relational structures was pioneeredby Frasse [64, 65, 66]. A relational structure R = (X, (Ri | i I)) onX is formed by the underlying set X and relations Ri Xmi on X; thesets X and I may be innite. The size of R is the cardinality of X andR is called nite (innite) if X is nite (innite). The signature of R isthe list (mi | i I) of arities mi N0 of the relations Ri . It is boundedif the numbers mi are bounded and is nite if I is nite.

Consider two relational structures R = (X, (Ri | i I)) and S =(Y, (Si | i I)) with the same signature and an injection f : X Ysatisfying, for every i in I and every mi-tuple (a1 , a2 , . . . , ami ) in X

mi ,

(a1 , a2 , . . . , ami ) Ri (f(a1), f(a2), . . . , f(ami )) Si.If such an injection f exists, we say that R is embeddable in S and writeR S. If in addition f is an identity (in particular, X Y ), R is asubstructure of S and we write R S. If the injection f is onto Y , wesay that R and S are isomorphic.

The age of a (typically innite) relational structure R on X is the setP of all nite substructures of R. Note that the age forms a downset inthe poset (U,) of all nite relational structures with the signature ofR whose underlying sets are subsets of X. The kernel of R is the set ofelements x in X such that the deletion of x changes the age. The proleof R is the unlabeled counting function gR (n) that counts nonisomorphicstructures with size n in the age of R. We get the same function if wereplace the age of R by the set P of all nite substructures embeddablein R whose underlying sets are [n]:

gR (n) = #({S = ([n], (Si | i I)) | S R}/)where is the isomorphism relation.


The next general result on growth of proles was obtained by Pouzet [95],see also [97].

Theorem 2.25 (Pouzet). If gR (n) is the prole of an innite relationalstructure R with bounded signature or with nite kernel, then exactlyone of the three cases occurs.

(i) The function gR (n) is constant for large n.(ii) There are constants k in N and 0 < c < d such that cnk nk for large n for every constant k in N.

Case 1 follows from the interesting fact that every innite R has a nonde-creasing prole (Pouzet [96], see [97, Theorem 4] for further discussion)and cases 2 and 3 were proved in [95] (see [97, Theorems 7 and 42]). It iseasy to see ([97, Theorem 10]) that for unbounded signature one can getarbitrarily slowly growing unbounded proles. Also, it turns out ([97,Fact 2]) that for bounded signature and nite-valued prole, one mayassume without loss of generality that the signature is nite. An innitegraph G = (N, E) whose components are cliques and that for every nhas innitely many components (cliques) of size n shows that for thesignature (2) the numbers of integer partitions pn appear as a prole(cf. Theorem 2.16)for relational structures in general (and unlabeledcount) there is no polynomial-exponential jump (but cf. Theorem 2.21).

The survey [97] contains, besides further results and problems on pro-les of relational structures, the following attractive conjecture whichwas partially resolved by Pouzet and Thiery [98].

Problem 2.26. In the cases 1 and 2 of Theorem 2.25 function gR (n) isa quasipolynomial for large n.

As remarked in [97], since gP (n) is nondecreasing, if the conjecture holdsthen the leading coecient in the quasipolynomial must be constant and,in cases 1 and 2, gR (n) = ank +O(nk1) for some constants a > 0 andk in N0 (cf. Theorem 2.16 and the following comment).

Relational structures are quite general in allowing arbitrarily many re-lations with arbitrary arities and therefore they can accommodate manypreviously discussed combinatorial structures and many more. On theother hand, ages of relational structures are less general than downsetsof structures, every age is a downset in the substructure ordering butnot vice versamany downsets of nite structures do not come from asingle innite structure (a theorem due to Frasse [97, Lemma 7], [44]

28 Klazar

characterizes downsets that are ages). An interesting research directionmay be to join the general sides of both approaches.

3 Four topics in general enumeration

In this section we review four topics in general enumeration. As we shallsee, there are connections to the results on growth of downsets presentedin the previous section.

3.1 Counting lattice points in polytopes

A polytope P in Rk is a convex hull of a nite set of points. If these pointshave rational, respectively integral, coordinates, we speak of rational,respectively lattice, polytope. For a polytope P and n in N we considerthe dilation nP = {nx | x P} of P and the number of lattice pointsin it,

fP (n) = #(nP Zk ).The following useful result was derived by Ehrhart [56] and Macdon-ald [83, 84].

Theorem 3.1 (Ehrhart, Macdonald). If P is a lattice polytope, respec-tively rational polytope, and fP (n) counts lattice points in the dilationnP , then there is a rational polynomial, respectively rational quasipoly-nomial, p(x) such that fP (n) = p(n) for every n.

For further renements and ramications of this result and its appli-cations see Beck and Robins [25] (also Stanley [111]). Barvinok [23]and Barvinok and Woods [24] developed a beautiful and powerful the-ory producing polynomial-time algorithms for counting lattice points inrational polytopes. In way of specializations one obtains from it manyWilan formulas. We will not say more on it because in its generality itis out of scope of this overview (as we said, this not a survey on #P).

3.2 Context-free languages

A language P is a subset of A, the innite set of nite words over anite alphabet A. The natural size function | | measures length of wordsand

fP (n) = #{u P | |u| = n}


is the number of words in P with length n. In this subsection thealphabet A is always nite, thus fP (n) |A|n .

We review the denition of context-free languages, for further infor-mation on (formal) languages see Salomaa [101]. A context-free grammaris a quadruple G = (A,B, c,D) where A,B are nite disjoint sets, c B(starting variable) and D (production rules) is a nite set of pairs (d, u)where d B and u (A B). A rightmost derivation of a wordv (A B) in G is a sequence of words v1 = c, v2 , . . . , vr = v in(A B) such that vi is obtained from vi1 by replacing the rightmostoccurrence of a letter d from B in vi1 by the word u, according to someproduction rule (d, u) D. (Note that no vi with i < r is in A.) We letL(G) denote the set of words in A that have rightmost derivation in G.If in addition every v in L(G) has a unique rightmost derivation in G,then G is an unambiguous context-free grammar. A language P A iscontext-free if P = L(G) for a context-free grammar G = (A,B, c,D). Pis, in addition, unambiguous if it can be generated by an unambiguouscontext-free grammar. If P is context-free but not unambiguous, we saythat P is inherently ambiguous. We associate with a context-free gram-mar G = (A,B, c,D) a digraph H(G) on the vertex set B by puttingan arrow d1 d2 , di B, if and only if there is a production rule(d1 , u) D such that d2 appears in u. We call a context-free languageergodic if it can be generated by a context-free grammar G such thatthe digraph H(G) is strongly connected.

Chomsky and Schutzenberger [49] obtained the following importantresult.

Theorem 3.2 (Chomsky and Schutzenberger). If P is an unambiguouscontext-free language and fP (n) counts words of length n in P , then thegenerating function

F (x) =n0

fP (n)xn

of P is algebraic over Q(x).The algebraicity of a power series F (x) =

n0 anx

n with an inN0 has two important practical corollaries for the counting sequence(an )n1 . First, as we already mentioned, it is holonomic. Second, it hasa nice asymptotics. More precisely, F (x) determines a function analyticin a neighborhood of 0 and if F (x) is not a polynomial, it has a niteradius of convergence , 0 < 1, and nitely many (dominating)singularities on the circle of convergence |x| = . In the case of single

30 Klazar

dominating singularity we have

an cnrn

where c > 0 is in R, r = 1/ 1 is an algebraic number, and the expo-nent is in Q\{1,2,3, . . . } (if F (x) is rational, then is in N0).For example, for Catalan numbers Cn = 1n+1

(2nn

)and their generating

function C(x) =

n0 Cnxn we have

xC2 C + 1 = 0 and Cn 1/2n3/24n .For more general results on asymptotics of coecients of algebraic powerseries see Flajolet and Sedgewick [63, Chapter VII].

Flajolet [62] used Theorem 3.2 to prove the inherent ambiguity ofcertain context-free languages. For further information on rational andalgebraic power series in enumeration and their relation to formal lan-guages see Barcucci et al. [22], Bousquet-Melou [36, 37], Flajolet andSedgewick [63] and Salomaa and Soittola [102].

How fast do context-free languages grow? Tromov [113] proved forthem a polynomial to exponential jump.

Theorem 3.3 (Tromov). If P A is a context-free language overthe alphabet A, then either fP (n) |A|nk for every n or fP (n) > cn forlarge n, where k > 0 and c > 1 are constants.

Tromov proved that in the former case in fact P w1w2 . . . wk for somek words wi in A. Later this theorem was independently rediscoveredby Incitti [73] and Bridson and Gilman [38]. DAlessandro, Intrigila andVarricchio [51] show that in the former case the function fP (n) is in facta quasipolynomial p(n) for large n (and that p(n) and the bound on ncan be eectively determined from P ).

Recall that for a language P A the growth constant is denedas c(P ) = lim sup fP (n)1/n . P is growth-sensitive if c(P ) > 1 andc(P Q) < c(P ) for every downset Q in (A,), where is the sub-word ordering, such that P Q = P . In other words, forbidding anyword u such that u v P for some v as a subword results in asignicant decrease in growth. Yet in other words, in growth-sensitivelanguages an analogue of MarcusTardos theorem (Theorem 2.2) holds.In a series of papers Ceccherini-Silberstein, Mach` and Scarabotti [46],Ceccherini-Silberstein and Woess [47, 48], and Ceccherini-Silberstein [45]the following theorem on growth-sensitivity was proved.

Theorem 3.4 (Ceccherini-Silberstein, Mach`, Scarabotti, Woess). Ev-


ery unambiguous context-free language P that is ergodic and has c(P ) >1 is growth-sensitive.

See [47] for extensions of the theorem to the ambiguous case and [46] forthe more elementary case of regular languages.

3.3 Exact counting of regular and other graphs

We consider nite simple graphs and, for a given set P N0 , the count-ing function (deg(v) = degG (v) is the degree of a vertex v in G, thenumber of incident edges)

fP (n) = #{G = ([n], E) | degG (v) P for every v in [n]}.For example, for P = {k} we count labeled k-regular graphs on [n]. Thenext general theorem was proved by Gessel [67, Corollary 11], by meansof symmetric functions in innitely many variables.

Theorem 3.5 (Gessel). If P is a nite subset of N0 and fP (n) countslabeled graphs on [n] with all degrees in P , then the sequence (fP (n))n1is holonomic.

This theorem was conjectured and partially proved for the k-regular case,k 4, by Goulden and Jackson [68]. As remarked in [67], the theoremholds also for graphs with multiple edges and/or loops. Domocos [54]extended it to 3-regular and 3-partite hypergraphs (and remarked thatGessels method works also for general k-regular and k-partite hyper-graphs). For more information see also Mishna [91, 90].

Consequently, the numbers of labeled graphs with degrees in xednite set have Wilan formula. Now we demonstrate this directly by amore generally applicable argument. For d in Nk+10 , we say that a graphG is a d-graph if |V (G)| = d0 + d1 + + dk , deg(v) k for every v inV (G) and exactly di vertices in V (G) have degree i. Let

p(d) = p(d0 , d1 , . . . , dk ) = #{G = ([n], E) | G is a d-graph}be the number of labeled d-graphs with vertices 1, 2, . . . , n = d0+ +dk .Proposition 3.6. For xed k, the list of numbers

(p(d) | d Nk+10 , d0 + d1 + + dk = m n)can be generated in time polynomial in n.

32 Klazar

Proof. A natural idea is to construct graphs G with di vertices of degreei by adding vertices 1, 2, . . . , n one by one, keeping track of the numbersdi . In the rst phase of the algorithm we construct an auxiliary (n+1)-partite graph

H = (V0 V1 Vn ,E)where we start with Vm consisting of all (k+1)-tuples d = (d0 , d1 , . . . , dk )in Nk+10 satisfying d0 + + dk = m and the edges will go only betweenVm and Vm+1. An edge joins d Vm with e Vm+1 if and only ifthere exist numbers 0 ,1 , . . . ,k1 in N0 such that: 0 i di for0 i k 1, r := 0 + +k1 k, andei = di+i1i for i {0, 1, . . . , k}\{r} but er = dr +r1r +1where we set 1 = k = 0. We omit from H (or better, do not con-struct at all) the vertices d in Vm not reachable from V0 = {(0, 0, . . . , 0)}by a path v0 , v1 , . . . , vm = d with vi in Vi . For example, the k verticesin V1 with d0 = 0 are omitted and only (1, 0, . . . , 0) remains. Also, welabel the edge {e, d} with the k-tuple = (0 , . . . ,k1). (It followsthat and r are uniquely determined by d, e.) The graph H togetherwith its labels can be constructed in time polynomial in n. It recordsthe changes of the numbers di of vertices with degree i caused by addingto G = ([m], E) new vertex m + 1; i are the numbers of neighbors ofm+ 1 with degree i in G and r is the degree of m+ 1.

In the second phase we evaluate a function p : V0 Vn Ndened on the vertices of H by this inductive rule: p(0, 0, . . . , 0) = 1 onV0 and, for e in Vm with m > 0,

p(e) =d

p(d)k1i=0

(dii

)where we sum over all d in Vm1 such that {d, e} E(H) and isthe label of the edge {d, e}. It is easy to see that all values of p canbe obtained in time polynomial in n and that p(d) for d in Vm is thenumber of labeled d-graphs on [m].

Now we can in time polynomial in n easily calculate

fP (n) =d

p(d)

as a sum over all d = (d0 , . . . , dk ) in Vn , k = maxP , satisfying di = 0when i P . Of course, this algorithm is much less eective than theholonomic recurrence ensured by (and eectively obtainable by the proof


of) Theorem 3.5. But by this approach we can get Wilan formula alsofor some innite sets of degrees P , for example when P is an arithmeticalprogression. (We leave to the reader as a nice exercise to count labeledgraphs with even degrees.) On the other hand, it seems to fail for manyclasses of graphs, for example, for triangle-free graphs.

Problem 3.7. Is there a Wilan formula for the number of labeledtriangle-free graphs on [n]? Can this number be calculated in time poly-nomial in n?

The problem of enumeration of labeled triangle-free graphs was men-tioned by Read in [99, Chapter 2.10]. A quarter century ago, Wilf [117]posed the following similar problem.

Problem 3.8. Can one calculate in time polynomial in n the numberof unlabeled graphs on [n]?

3.4 Ultimate modular periodicity

One general aspect of counting functions fP (n) not touched so far istheir modular behavior. For given modulus m in N, what can be saidabout the sequence of residues (fP (n) mod m)n1 . Before presenting arather general result in this area, we motivate it by two examples.

First, Bell numbers Bn counting partitions of [n]. Recall thatn0

Bnxn =

k=0

xk

(1 x)(1 2x) . . . (1 kx) .

Reducing modulo m we get, denoting v(x) = (1x)(12x) . . . (1(m1)x),

n0Bnx

n mj=0

xmj

v(x)j

m1i=0

xi

(1 x)(1 2x) . . . (1 ix)

=1

1 xm/v(x)m1i=0

xi

(1 x)(1 2x) . . . (1 ix)

=a(x)

v(x) xm =a(x)

1 + b1x+ + bmxm=

n0

cnxn

where a(x) Z[x] has degree at most m1 and bi are integers, bm = 1.Thus the sequence of integers (cn )n1 satises for n > m the linear

34 Klazar

recurrence cn = b1cn1 bm cnm of order m. By the pigeonholeprinciple, the sequence (cn mod m)n1 is periodic for large n. SinceBn cn mod m, the sequence (Bn mod m)n1 is periodic for large n aswell. (For modular periods of Bell numbers see Lunnon, Pleasants andStephens [82]).

Second, Catalan numbers Cn = 1n+1(2n

n

)counting, for example, non-

crossing partitions of [n]. The shifted version Dn = Cn1 = 1n(2n2

n1)

satises the recurrence D1 = 1 and, for n > 1,

Dn =n1i=1

DiDni = 2n/21

i=1

DiDni +

n/2

i=n/2DiDni .

Thus, modulo 2, D1 1, Dn 0 for odd n > 1 and Dn D2n/2 Dn/2for even n. It follows that Dn 1 if and only if n = 2m and thatthe sequence (Cn mod 2)n1 has 1s for n = 2m 1 and 0s elsewhere.In particular, it is not periodic for large n. (For modular behavior ofCatalan numbers see Deutsch and Sagan [53] and Eu, Liu and Yeh [58]).

Bell numbers come out as a special case of a general setting. Considera relational system which is a set P of relational structures R with thesame nite signature and underlying sets [n] for n ranging in N. We saythat P is denable in MSOL, monadic second-order logic, if P coincideswith the set of nite models (on sets [n]) of a closed formula in MSOL.(MSOL has, in addition to the language of the rst-order logic, variablesS for sets of elements, which can be quantied by ,, and atomicformulas of the type x S; see Ebbinghaus and Flum [55].) Let fP (n)be the number of relational structures in P on the set [n], that is, thenumber of models of on [n] when P is dened by . For example, the(rst-order) formula given by (a, b, c are variables for elements, is abinary predicate)

a, b, c : (a a) & (a b b a) & ((a b & b c) a c)has as its models equivalence relations and fP (n) = f(n) = Bn , theBell numbers.

Let us call a sequence (s1 , s2 , . . . ) ultimately periodic if it is periodicfor large n: there are constants p, q in N such that sn+p = sn whenevern q. Specker and Blatter [27, 28, 29] (see also Specker [107]) provedthe following remarkable general theorem.

Theorem 3.9 (Specker and Blatter). If a relational system P den-able in MSOL uses only unary and binary relations and fP (n) counts


its members on the set [n], then the sequence (fP (n) mod m)n1 is ul-timately periodic for every m N .The general reason for ultimate modular periodicity in Theorem 3.9 isthe same as in our example with Bn , residues satisfy a linear recurrencewith constant coecients. Fischer [59] constructed counterexamples tothe theorem for quaternary relations, see also Specker [108]. Fischer andMakowski [60] extended Theorem 3.9 to CMSOL (monadic second-orderlogic with modular counting) and to relations with higher arities whenvertices have bounded degrees. Note that regular graphs and triangle-free graphs are rst-order denable. Thus the counting sequences men-tioned in Theorem 3.5 and in Problem 3.7 are ultimately periodic to anymodulus. More generally, any hereditary graph property P = Av(F )with nite base F is rst-order denable and similarly for other struc-tures. Many hereditary properties with innite bases (and also many setsof graphs or structures which are not hereditary) are MSOL-denable;this is the case, for example, for forests (P = Av(F ) where F is theset of cycles) and for planar graphs (use Kuratowskis theorem). To allof them Theorem 3.9 applies. On the other hand, as the example withCatalan numbers shows, counting of ordered structures is in general outof reach of Theorem 3.9.

A closely related circle of problems is the determination of spectra ofrelational systems P and more generally of nite models; the spectrumof P is the set

{n N | fP (n) > 0}the set of sizes of members of P . In several situation it was provedthat the spectrum is an ultimately periodic subset of N. See Fischer andMakowski [61] (and the references therein), Shelah [104] and Shelah andDoron [105]. We conclude with a problem posed in [59].

Problem 3.10. Does Theorem 3.9 hold for relational systems withternary relations?

Acknowledgments My thanks go to the organizers of the conferencePermutations Patterns 2007 in St AndrewsNik Ruskuc, Lynn Hynd,Steve Linton, Vince Vatter, Miklos Bona, Einar Steingrmsson and Ju-lian Westfor a very nice conference which gave me opportunity andincentive to write this overview article. I thank also Vt Jelnek forreading the manuscript and an anonymous referee for making many im-provements and corrections in my style and grammar.

36 References

References[1] M. H. Albert and M. D. Atkinson. Simple permutations and pattern restricted

permutations. Discrete Math., 300(1-3):115, 2005.[2] M. H. Albert, M. D. Atkinson, and R. Brignall. Permutation classes of poly-

nomial growth. Ann. Comb., 11(34):249264, 2007.[3] M. H. Albert, M. Elder, A. Rechnitzer, P. Westcott, and M. Zabrocki. On the

Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia.Adv. in Appl. Math., 36(2):95105, 2006.

[4] M. H. Albert and S. Linton. Growing at a perfect speed. Combin. Probab.C

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