Boolean Lattice and Symmetric Chain Decompositionsmath.sjtu.edu.cn/conference/Bannai/2014/data/20141230B/slides.pdf · (1) An SCO poset P is necessarily Peck. (2) Any quotient of

Boolean Lattice and Symmetric Chain Decompositions

Yizhe Zhu

Shanghai Jiao Tong University

[email protected]

December 30, 2014

Yizhe Zhu (SJTU) Symmetric Chain Decomposition December 30, 2014 1 / 47

Overview

1 Boolean LatticeDefinitionParenthesis MatchingConditions for an SCO

2 Necklace PosetBlock CodeModified Parenthesis MatchingLyndon RearrangementGeneralization

3 ApplicationsSymmetric Venn DiagramsSymmetric Independent Families

4 Open Problems


Symmetric Chain Decomposition (SCD)

Let (P, <) be a finite poset. A chain in P is a sequencex1 < x2 < ... < xn, where each xi ∈ P. For x , y ∈ P, we say y covers x ifx < y and there does not exist z ∈ P such that x < z and z < y .A saturated chain in P is a chain where each element is covered by thenext.P is ranked if there exists a function r : P → Z≥0 such that x covers yimplies r(y) = r(x) + 1. Suppose min{r(x)|x ∈ P} = 0, the rank of P isdenoted r(P) = max{r(x)|x ∈ P}.A saturated chain x1 < x2 < ... < xn in a ranked poset P is said to besymmetric if r(x1) + r(xn) = r(P).P has a symmetric chain deomposition (SCD) if it can be written as adisjoint union of saturated, symmetric chains.A symmetric chain order (SCO) is a finite ranked poset for which thereexists a symmetric chain decomposition.


Boolean Lattice

The Boolean lattice, denoted Bn, is the power set of [n] = {1, 2, ..., n}ordered by inclusion. r(A) = |A| for all A ⊂ [n]. An element A ∈ Bn canbe viewed as an n-bit binary string whose ith bit is 1 if i ∈ A, 0 if i 6∈ A.


Parenthesis Matching

parenthsis matching: start at the left, when a 1 is encounterd, it ismatched to the rightmost unmatched zero (if any) with a pair of brackets.Continue in this manner until we reach the end of the string.

x = 1011011100010110 the parentesis: 1(01)1(01)110(0(01)(01)1)0.U0(x) = {9, 16} unmatched zerosU1(x) = {1, 4, 7, 8} unmatched onesM(x) = {(2, 3), (5, 6), (10, 15), (11, 12), (13, 14)} matched pairs

τ : change the leftmost unmatched 1 to a 0, defined on x ∈ Bn,U0(x) 6= ∅.


Parenthesis Matching

Theorem (Greene, Kleitman, 1976)

For x ∈ Bn with |U0(x)| = k , let Cx = {x , τ(x), τ2(x), ..., τk(x)}. Thefollowing is a symmetric chain decomposition of Bn:

S = {Cx |x ∈ Bn,U1(x) = ∅}

Figure: the SCD of B4 by parenthesis matching


Which posets are SCO?

Given a ranked poset P, r(P) = M. Pk = {x ∈ P|r(x) = k}.rank-symmetric: |Pk | = |PM−k |.rank-unimodal: there exists j such that |P0| ≤ |P1| ≤ ... ≤ |Pj | and|Pj | ≥ |Pj+1| ≥ ... ≥ |PM |.antichain: a set of pairwise uncomparable elements of a posetstrongly Sperner: for all k = 1, 2, ...,M + 1,the union of the k middlelevels of P is a unition of k antichains of maximum size.A poset is Peck if it is rank-symmetric, rank-unimodal, and stronglySperner.

Given a group G of automorphisms of a poset P, a quotient poset of Punder G , denoted P/G , is orderd in the following way:For orbits of G , A and B, we have A ≤ B iff there are a ∈ A, b ∈ B suchthat a ≤ b in P.


Which posets are SCO?

Stanley gave the necessary condition for an SCO.

Theorem (Stanley, 1984)

(1) An SCO poset P is necessarily Peck.(2) Any quotient of the Boolean lattice is a Peck poset.

Griggs showed a sufficient condition for an SCO.

Theorem (Griggs, 1977)

LYM property, rank-symmetry and rank-unimodality implies that a poset Pis SCO.

LYM property: for every antichain F ,∑

x∈F1

|Pr(x)|≤ 1.


Necklace Poset

rotation σ on Bn: x = (x1, x2, ..., xn), σ(x) = (xn, x1, ..., xn−1).For x , y ∈ Bn, we say x ∼ y if y = σk(x) for some k.The necklace poset Nn is the quotient poset of Bn under the equivalencerelation ∼. For X ,Y ∈ Nn, X ≤ Y iff there exist x ∈ X , y ∈ Y , x ≤ y .

For prime p, it is known that Np satisfies the LYM property and has anSCD. For general n, LYM property is unknown.


Block Codes

The explicit construction of an SCD for Np was given by the idea of blockcode together with parenthesis matching. A surprising application of suchSCD of Np is to construct symmetric Venn diagrams.

Theorem (Griggs, Killian, Savage, 2004)

For prime n, there is a way to select a set Rn consisting of onerepresentative from each necklace in Nn such that the inducednecklace-representative subposet (Rn,≤) of Bn has an SCD.

It is a stronger than the claim that Np has an SCD.


Block Codes

block code β(x) of a binary string x :If x starts with 0 or ends with 1, β(x) = (∞).If x = 1a10b11a20b2 ...1at 0bt , where t > 0, ai > 0, bi > 0, 1 ≤ i ≤ t, thenβ(x) = (a1 + b1, a2 + b2, ..., at + bt).Consider the rotation of x = 0011011, we haveβ(0011011) = (∞), β(0110110) = (∞), β(1101100) = (3, 4),β(1011001) = (∞), β(0110011) = (∞), β(1100110) = (4, 3),β(1001101) = (∞).

Lemma (Griggs, Killian, Savage, 2004)

If n is prime, no two strings of {0, 1}n in the same necklace have the samefinite block code.

The representative ρ(x) of necklace containing x is the rotation y of x forwhich β(y) is minimum. By the lemma, ρ(x) is the unique representative.For example, ρ(0011011) = 1101100.


Block Codes

Rn = {ρ(x)|x ∈ {0, 1}n} = {x ∈ {0, 1}n|ρ(x) = x} is a subposet of Bn,called necklace-representative poset.Consider R∗n = Rn − {0n, 1n}, we construct an SCD for Rn by showing R∗nhas an SCD.

Lemma (Griggs, Killian, Savage, 2004)

If x ∈ R∗n and |U0(x)| ≥ 2, then τ(x) ∈ R∗n . Similarly, if |U1(x)| ≥ 2, thenτ−1(x) ∈ R∗n .

Define the set S∗ = {z ∈ R∗n |U1(z) = {1}}. For z ∈ S∗, define the chainof z by Jz = z , τ(z), τ2(z), ...τk−1(z).Every x ∈ R∗n is in the chain Jz for some z ∈ S∗. The set of chains{Jz |z ∈ S∗} is an SCD of R∗n .Extend {Jz |z ∈ S∗} to an SCD in Rn by extending the chain10n−1 < 110n−2 < ... < 1n−200 < 1n−10 to the chain0n < 10n−1 < 110n−2 < ... < 1n−200 < 1n−10, 1n.



Is Nn an SCO for all n ≥ 1?

Theorem (Jiang, Savage, 2009)

Nn is an SCO for all prime n and for composite n ≤ 18.

They considered the periodic block code but failed to generalize theconstruction for lager n.

Theorem (Jordan, 2010)

The necklace poset is a symmetric order for all n ≥ 1.

The proof is constructive and the most important step is to choose therepresentative of each necklace.


Modified Parenthesis Matching

Consider a set Mn consisting of x ∈ Bn such that x achieves the maximumnumber of unmatched ones over all rotations.

Mn = {y ∈ Bn : |U1(y)| = max{|U1(σk(y))| : k = 1, 2, ..., n}}

Figure: the SCD for B6 with memebers of M6 in bold


Lemma (Jordan, 2010)

Let x ∈ Mn.(1)If |x | < n

2 , τ i (x) ∈ Mn, 1 ≤ i ≤ n − 2|x |.(2) If |x | > n

2 , τ−i (x) ∈ Mn, 1 ≤ i ≤ 2|x | − n.


Modified Parenthesis Matching

If x ∈ Mn and Cx is the chain containing x in the SCD of Bn, the smallestsymmetric subchain of Cx containing x is also in Mn. Note that theresulting chains still contain at least one representative of each necklace.

Figure: the SCD for M6 with duplicate representatives of N6



Let x , y ∈ Mn with x ∼ y .(1) If |x | ≥ n

2 , then τ(x) ∼ τ(y) or {τ(x), τ(y)} ∩Mn = ∅.(2) If |x | ≤ n

2 , then τ−1(x) ∼ τ−1(y) or {τ−1(x), τ−1(y)} ∩Mn = ∅.


Let x , y ∈ Bn with |x | = |y | ≤ n2 , then x ∼ y ⇔ τn−2k(x) ∼ τn−2k(y)

The three lemmas above are proven by the idea of circular matching.


Remove duplicate representatives in Mn

C 0x : the chain containing x in the Greene-Kleitman SCD for Bn restricted

to Mn. D0 = Mn.Before step j + 1 in the iteration, C j

x : the chain containing x . D j : the setof elements of Mn remaining in the poset.

For step j + 1, let x 6= y ∈ D j with x ∼ y , |x | = k. If there are no suchx , y , we have an SCD for Nn.Suppose |x | ≤ n

2 , and C jx is at least as long as C j

y .

bottom tail T jb = {τ−i (y)|i ≥ 0, τ−i (y) ∈ Mn}.

top tail T jt = {τn−2k+i (y)|i ≥ 0, τn−2k+i (y) ∈ Mn}.

C j+1∗ = C j

y \ (T jb ∪ T j

t ) D j+1 = D j \ (T jb ∪ T j

t ).

For z ∈ D j+1 \ C jy , C j+1

z = C jz .

For z ∈ C j+1∗ ,C j+1

z = C j+1∗ .

Then D j+1 ( D j and⋃

z∈D j+1 Cj+1z is an SCD for D j+1.


Figure: Removing duplicate representatives of elements of Mn


Figure: the resulting SCD for N6


Lyndon Rearrangement

Hersh and Schilling gave an explicit construction of SCD of Nn for generaln by the idea of Lyndon words. The resulting symmetric chains aredifferent from the previous work by Jordan.

Given an ordered alphabet A and a word w ∈ An, define the Lyndonrearrangement of w to be the lexicographically smallest word obtained bya rotation of the letters in w . The resulting word is called a Lyndon word.

For Bn, A = {0, 1} with ordering 1 ≺ 0.Lyndon rearrangement of 0001110 is 1110000.


Define the Map φ

On the top half (|x | ≥ n2 ) of Nn, given a word w .

(1) Cyclically rotate it into its Lyndon rearrangement.(2) Add brackets within the word by the following procedure:

(i) Take any 0 that is immediately followed by 1 cyclically and match thesepairs by brackets.(ii) Remove them from further consideration. Follow the processrepeatedly until all unmatched elements are 1’s.

For example:1101100110→ 1)1(01)10(01)1(0→ 1)1(01)1(0(01)1)(0. We call these 01pairs a matching pair.


Define the Map φ

If there is at least one unpaired 1 at the end of this process, then φ mapsthe word to a word in which the rightmost unpaired 1 in Lyndonexpression is changed to 0.

φ maps 1)1(01)1(0(01)1)(0 to 1101000110.The image of φ in Lyndon rearrangment is 1101101000 with brackets1)1)(01)1)(01)(0(0(0.

Define φ on the lower half ( |x | < n2 ) of Nn by successively undoing the

most recently created matching pair by turning its letter 1 to a 0.


Define the Map φ

Example

Repeated application of φ to 111101100101111000 yields a symmetricchain.1)1)1)1)(01)1(0(01)(01)1)11(0(0(0(0→1)1)1)1)(01)1)(0(01)(01)1)1(0(0(0(0(0→1)1)1)1)(01)1)(0(01)(01)1)0(0(0(0(0(0→1)1)1)1)(01)0(0(01)(01)1)00(0(0(0(0

Theorem (Hersh, Schilling, 2013)

The necklace poset Nn has an SCD such that u, v ∈ Nn with |u| < |v |belong to the same chain if and only if u = φr (v) for some r > 0.


Apply the Map φ

With the application of φ we get the SCD for N5.


The map φ is exactly the Kashiwara lowering operator of a cyclic crystal.


Generalization

The necklace poset Nn is the quotient of Bn under the action of the cyclicgroup Zn. Other results about the quotients of Bn and general poset P areas follows.

Theorem (Duffus, Mckibben-Sanders, Thayer, 2012)

(1)Let G be a subgroup of Sn generated by powers of disjoint cycles. Thenthe poset Bn/G is an SCO.(2)Let G be a 2-element subgroup with non-unit element a product ofdisjoint transpositions. Then the poset Bn/G is an SCO.(3)Let C be a chain and let K be a subgroup of Sm generated by powersof disjoint cycles. Then Cm/K is an SCO.

Theorem (Dhand, 2012)

Let P be a poset. If P is an SCO, then Pn/Zn is an SCO.


An n-Venn diagram is a collection of n simple closed curves in the plane,{Θ1,Θ2, ...,Θn} such that for each S ⊂ {1, 2, ..., n} the region⋂

i∈S int(Θi ) ∩⋂

i 6∈S ext(Θi ) is nonempty and connected. A Venn diagramis called simple if each point of intersection has degree 4.


Symmetric Venn Diagrams

A symmetric Venn diagram is one with rotational symmetry. That is, thereis a point p in the plance such that each of the n rotations of Θ1 about pby an angle of 2πi/n, 0 ≤ i ≤ n − 1, coincides with one of the curvesΘ1,Θ2, ...,Θn.

Theorem (Handerson, 1963)

Symmetric Venn diagrams are not possible when n is not prime.

Theorem (Griggs, Killian, Savage, 2004)

For all prime n, there is a symmetric n-Venn diagram which can beconstructed from the SCD in Rn.


Let C be an SCD in a poset A. Call the longest chains in C the rootchains. C has the chain cover propety if whenever C ∈ C and C is not aroot chain, there exists a chain π(C ) ∈ C such that starter(C ) covers anelement πs(C ) of π(C ), terminator(C ) is covered an element πt(C ) ofπ(C ). Call such a map π a chain cover map.The chain cover graphG (C, π) is shown.








Griggs, Killian and Savage asked in their paper: does there always exist asimple symmetric n-Venn diagram when n is prime?Examples are known only for n = 3, 5, 7.A simple symmetric Venn diagram contains 2n − 2 intersection points.Killian, Ruskey, Savage and Weston (2004) showed the method of Griggs,Killian and Savage failed to find simple symmetric Venn diagrams.They constructed the half-simple symmetric Venn diagrams: symmetricVenn diagrams with asympotically at least 2n−1 intersection points.



To date the simplest symmetric 11-Venn diagram is due to Hamburger,Petruska and Sali. Their diagram has 1837 vertices and is about 90%simple.


An independent family is a collection of n curves in the plane such thatevery subset of [n] is represented at least once in the regions formed by theintersection of the interiors of the curves. A Venn diagram is anindependent family where each subset is represented once.

Theorem (Grunbaum, 1999)

(1)Any independent family of n curves must have at least 2 + n(|Nn| − 2)regions.(2) symmetric independent families of n curves exists for all n.

Theorem (Jiang, 2003)

If there exists an SCD of Rn with chain cover property, then there exits asymmetric independent family of n curves with 2 + n(|Nn| − 2) regions.

Theorem (Jordan, 2010)

There exits a symmetric independent family of n curves with2 + n(|Nn| − 2) regions.


Open Problems

(Griggs, Killian, Savage, 2004) Construct simple symmetric n-Venndiagrams for prime n ≥ 11.

(Canfield, Mason, 2006) For all subgroups G of Sn, is Bn/G an SCO?

(Jordan, 2010) Consider the true necklace, meaning Bn/G , where Gis the group of automorphism that includes both rotations andinversions. Does true necklace have an SCD?

(Jordan, 2010) Let G and H be two groups of automorphisms on Bn,and K the group of automorphisms generated by G and H. If Bn/Gand Bn/H are SCOs, is Bn/K also an SCO?

SCD for L(m, n), i.e. poset of partitions in a rectangle.

SCD for poset Sn with weak order or Bruhat order.


References

Canfield, E. R., & Mason, S. (2006).

When is a quotient of the Boolean lattice a symmetric chain order

Preprint.

Douglas B. W. (1980).

A symmetric chain decomposition of L(4, n)

European Journal of Combinatorics, 1(4), 379-383.

Duffus, D., McKibben-Sanders, J., & Thayer, K. (2012).

Some quotients of chain products are symmetric chain orders

Electronic Journal of Combinatorics, 19(2), 46-57.

Greene, C., & Kleitman, D. J. (1976).

Strong versions of Sperner’s theorem

Journal of Combinatorial Theory, Series A, 20(1), 80-88.

Griggs, J. R. (1977).

Sufficient conditions for a symmetric chain order

SIAM Journal on Applied Mathematics, 32(4), 807-809.


References

Griggs, J., Killian, C. E., & Savage, C. D. (2004).

Venn diagrams and symmetric chain decompositions in the Boolean lattice

Electronic Journal Combinatorics, 11(1).

Grunbaum, B. (1975).

Venn diagrams and independent families of sets

Mathematics Magazine, 12-23.

Grunbaum, B. (1999).

The search for symmetric Venn diagrams

Geombinatorics, 8(4), 104-109.

Hamburger, P., Petruska, G., & Sali, A. (2004).

Saturated chain partitions in ranked partially ordered sets, and non-monotonesymmetric 11-Venn diagrams

Studia Scientiarum Mathematicarum Hungarica, 41(2), 147-192.

Henderson, D. W. (1963).

Venn diagrams for more than four classes

American Mathematical Monthly, 424-426.


References

Hersh, P., & Schilling, A. (2013).

Symmetric chain decomposition for cyclic quotients of Boolean algebras andrelation to cyclic crystals

International Mathematics Research Notices, 2013(2), 463-473.

Jiang, Z. (2003).

Symmetric chain decompositions and independent families of curves

MS thesis, North Carolina State University.

Jiang, Z., & Savage, C. D. (2009).

On the existence of symmetric chain decompositions in a quotient of the Booleanlattice

Discrete Mathematics, 309(17), 5278-5283.

Jordan, K. K. (2010).

The necklace poset is a symmetric chain order

Journal of Combinatorial Theory, Series A, 117(6), 625-641.

Killian, C. E., Ruskey, F., Savage, C. D., & Weston, M. (2004).

Half-simple symmetric Venn diagrams

Journal of Combinatorics, 11(3), R86.Yizhe Zhu (SJTU) Symmetric Chain Decomposition December 30, 2014 45 / 47

References

Ruskey, F., & Weston, M. (1997).

A survey of Venn diagrams

Electronic Journal of Combinatorics, 4.

Ruskey, F., Savage, C. D., & Wagon, S. (2006).

The search for simple symmetric Venn diagrams

Notices of the AMS, 53(11), 1304-1311.

Stanley, R. P. (1984).

Quotients of Peck posets

Order, 1(1), 29-34.

Stanley, R. P. (2011).

Enumerative combinatorics (Vol. 1)

Cambridge University Press.

Vivek, D. (2012).

Symmetric chain decomposition of necklace posets

Electronic Journal of Combinatorics, 19(1), 26-40.


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Boolean Lattice and Symmetric Chain Decompositionsmath.sjtu.edu.cn/conference/Bannai/2014/data/20141230B/slides.pdf · (1) An SCO poset P is necessarily Peck. (2) Any quotient of