Research Statement for Brian K. Miceli

My mathematical interests lie mainly in the field of combinatorics and how it relates to other fieldsof mathematics. My current research is in the area of enumerative and algebraic combinatorics withan emphasis on rook theory and Stirling numbers. Within the realm of combinatorics there are agreat number of problems to work on which appeal to those of all levels of mathematical maturity.This leads to ample possibility of both graduate and undergraduate level research.

This document will describe some of the main results from the research that I have been con-ducting for the last two years at the University of California, San Diego, as well as discussing theareas in which there is opportunity for further study and research.

Classical Rook Theory, Q-Analogues, & Product Formulas.

Let N = {1, 2, 3, . . .} denote the set of natural numbers. For any positive integer a we will set[a] := {1, 2, . . . , a}. We will say that Bn = [n]× [n] is an n-by-n array of squares (like a chess board),which we call cells. The cells of Bn will be numbered from left to right and bottom to top with thenumbers from [n], and we will refer to the cell in the ith row and jth column of Bn as the (i, j) cell ofBn. Any subset of Bn is called a board, and if B is a board in Bn with column heights b1, b2, . . . , bn,with 0 ≤ bi ≤ n for each i, then we will write B = F (b1, b2, . . . , bn) ⊆ Bn. In the special case that0 ≤ b1 ≤ b2 ≤ · · · ≤ bn ≤ n, we will say that B = F (b1, b2, . . . , bn) is a Ferrers board.

Given a board B = F (b1, b2, . . . , bn), there are three sets of numbers we can associate with B,namely, the rook, file, and hit numbers of B. The rook number, rk(B), is the number of placementsof k rooks in the board B so that no two rooks lie in the same row or column. The file number,fk(B), is the number of placements of k rooks in the board B so that no two rooks lie in the samerow. Given a permutation, σ = σ1σ2 . . . σn, in the symmetric group Sn, we shall identify σ with theplacement Pσ = {(1, σ1), (2, σ2), . . . , (n, σn)}. Then the hit number hk(B) is the number of σ ∈ Sn

such that the placement Pσ intersects the board in exactly k cells.All of these numbers have been studied extensively by combinatorialists, and here are three

fundamental identities involving these numbers:n∑

k=0

hk(B)(x + 1)k =n∑

k=0

rk(B)(n− k)!xk, (1)

n∏

i=1

(x + bi − (i− 1)) =n∑

k=0

rn−k(B)(x) ↓k, and (2)

n∏

i=1

(x + bi) =n∑

k=0

fn−k(B)xk. (3)

where (x) ↑m= x(x + 1) · · · (x + (m− 1)) and (x) ↓m= x(x− 1) · · · (x− (m− 1)). Identity (1) is dueto Kaplansky and Riordan [10] and holds for any board B ⊆ Bn. Identity (2) holds for all Ferrersboards B = F (b1, . . . , bn) and is due to Goldman, Joichi, and White [6]. Identity (3) is due to Garsiaand Remmel [3] and holds for all boards of the form B = F (b1, . . . , bn).

We note that in the special case where B := Bn = F (0, 1, 2, . . . , n − 1) Equations (2) and (3)become

xn =n∑

k=0

rn−k(Bn)(x) ↓k and (4)

(x) ↑k=n∑

k=0

fn−k(Bn)xk, (5)

which show that rn−k(Bn) = Sn,k, where Sn,k is the Stirling number of the second kind, and(−1)n−kfn−k(Bn) = sn,k, where sn,k is the Stirling number of the first kind. Thus, we have naturalrook theory interpretations for the Stirling numbers of the first and second kind.

There are natural q-analogues of formulas (1), (2), and (3). Define [n]q = 1 + q + · · · + qn−1 =1− qn

1− q. Then we also define

[n]q! = [n]q[n− 1]q · · · [2]q[1]q, (6)[x]q ↓m = [x]q[x− 1]q · · · [x− (m− 1)]q, and (7)[

n

k

]

q

=[n]q!

[k]q![n− k]q!. (8)

Garsia and Remmel [3] defined q-analogues of the hit numbers, hk(B, q), q-analogues of the rooknumbers, rk(B, q), and q-analogues of file numbers fk(B, q) for Ferrers boards B so that the followinghold:

n∑

k=0

hk(B, q)xn−k =n∑

k=0

rn−k(B, q)[k]q!xk(1− xqk+1) · · · (1− xqn), (9)

n∏

i=1

[x + bi − (i− 1)]q =n∑

k=0

rn−k(B, q)[x]q ↓k, and (10)

n∏

i=1

[x + bi]q =n∑

k=0

fn−k(B, q)[x]k. (11)

In recent years, a number of researchers have developed new rook theory models which give riseto new classes of product formulas. For example, Haglund and Remmel [8] developed a rook theorymodel where the analogue of the the rook number, mk(B), counts partial matchings in the completegraph Kn. They defined an analogue a Ferrers board B and in their setting and proved the followingidentity,

2n−1∏

i=1

(x + a2n−i − 2i + 2) =2n−1∑

k=0

mk(F )x(x− 2)(x− 4) · · · (x− 2(2n− 1− k)). (12)

Remmel and Wachs [13] defined a more restricted class of rook numbers, rjk(B), and an analogue of

Ferrers boards called j-attacking Ferrers boards. They proved the following identity:

n∏

i=1

(x + bi − j(i− 1)) =n∑

k=0

rjn−k(B)x(x− j)(x− 2j) · · · (x− (k − 1)j). (13)

Goldman and Haglund [5] developed an i-creation rook theory model and proved that for Ferrersboards one has the following identity:

n∏

j=1

(x + bi + j(i− 1)) =n∑

k=0

r(i)n−k(B)x(x + (i− 1)) · · · (x + (k − 1)(i− 1)). (14)

In all of these models, one can prove q-analogues of these of product formulas.

2

General Augmented Rook Boards.

A natural question arises if one can give a uniform explanation of all these products formulas.The first part of my thesis develops a completely new rook theory model where one can prove aquite general product formula which includes all the product formulas described above. One specialcase of our general product formula is the following. Let a = (a1, a2, . . .) be any sequence of naturalnumbers and let A = (A1, A2, . . . , An) be the sequence of partial sums where Ai = a1 + . . . + ai. Wedefine a new class of rook numbers rA

k (B) and rAk (B) for any board B = F (b1, . . . , bn) such that the

following product formulas hold:

n∏

i=1

(x + bi) =n∑

k=0

rAk (B)x(x−A1)(x−A2) · · · (x−Ak−1) and (15)

n∏

i=1

(x + bi) =n∑

k=0

rAk (B)x(x + A1)(x + A2) · · · (x + Ak−1). (16)

The goal of the first part of my thesis was to develop a single rook theory model to describe bothof the above formulas. To give a flavor of the type of rook placements that we consider, I will definethe rook numbers rA

k (B) and rAk (B). I first need to define the augmented rook board, BA, which is

the rook board with column heights (from left to right) b1 +A1, b2 +A2, . . ., bn +An. We will call thepart of BA that corresponds to the Ai’s the augmented part of BA, and we will define cancelation inBA as follows: a rook placed in column j of BA will cancel, in each column, all of the cells to its rightwhich lie in the ai of highest subscript. An example of these boards, along with the correspondingcancelation, can be seen in Figure 1, with B = F (1, 3, 3, 4) and a = (1, 2, 2, 1). Here, cancelation bythe rook in the first column is denoted by “•” and the cancelation from the rook in the third columnis denoted by “∗”. In performing this cancelation, we define rA

k (B) to be the number of ways ofplacing k rooks in BA. Similarly, if we consider the exact same placements, but weight every rookwhich is placed in the augmented part of BA with a “−1”, then we will let rA

k (B) denote the sum ofall such weighted placements of k rooks in BA. One can then define placements of rooks on a generalaugmented rook board, which is a completely different board than any previous models, to generatethe product formulas in (15) and (16). I can also prove q-analogues of Equations (15) and (16).Further Study:Are there generalizations of Laguerre boards and corresponding ideas of orthogonalpolynomials studied by Gessel [4]?

Poly-Stirling Numbers.

Classical Stirling numbers satisfy the recursions

Sn+1,k = Sn,k−1 + kSn,k (17)

andsn+1,k = sn,k−1 − nsn,k, (18)

where S0,0 = s0,0 = 1, and Sn,k = sn,k = 0 whenever n < 0, k < 0, or n < k. The numbers Sn,k arereferred to as the Stirling numbers of the second kind and the numbers sn,k are referred to as theStirling numbers of the first kind. It is well known that if ||Sn,k|| and ||sn,k|| are the matrices formedfrom the above numbers, then ||Sn,k|| = ||sn,k||−1.

There have been many generalizations of Stirling numbers studied in the literature (see Gould[7], Hsu and Shiue [9], de Medicis and Leroux [11], Wachs [16], Wachs and White [17]). In the second

3

2 3 4

4

X

X

a

a

aa

a

aa

aa

1

.....**

1

1 1

12 2

2

3

3

abbbb

Figure 1: An example of the board BA and the corresponding cancelation with B = F (1, 3, 3, 4) anda = (1, 2, 2, 1).

part of my thesis I study what I call Poly-Stirling Numbers, which are defined as follows: let p(x) beany polynomial with coefficients in N ∪ {0}, and consider the numbers S

p(x)n,k and s

p(x)n,k defined by

Sp(x)n+1,k = S

p(x)n,k−1 + p(k)Sp(x)

n,k (19)

andsp(x)n+1,k = s

p(x)n,k−1 − p(n)sp(x)

n,k , (20)

where Sp(x)0,0 = s

p(x)0,0 = 1, and S

p(x)n,k = s

p(x)n,k = 0 whenever n < 0, k < 0, or n < k.

I call the numbers in Equation (19) the Poly-Stirling numbers of the second kind and the numbersin Equation (20) the Poly-Stirling numbers of the first kind. I also define c

p(x)n,k = (−1)n−ks

p(x)n,k to be

the signless Poly-Stirling numbers of the first kind, and I can give rook theory interpretations Sp(x)n,k

and cp(x)n,k . Another result, which follows from Milne Inversion [12], is that ||Sp(x)

n,k || = ||sp(x)n,k ||−1. I

also have a bijective, combinatorial proof of this fact.A special class of these numbers, which I call xm-Stirling numbers, is when p(x) = xm for some

m ∈ N ∪ {0}. In the case of m = 1, these are the regular Stirling numbers, and when m = 2,these numbers are referred to as Central Factorial Numbers in both Riordan [14] and Stanley [15].The following results pertaining to xm-Stirling numbers were obtained through my research. All ofthese results have completely combinatorial proofs, and they can all be generalized for Poly-Stirlingnumbers:

n∏

i=1

(xm + (i− 1)m) =n∑

k=0

cxm

n,k(xm)k, (21)

n∏

i=1

(xm − (i− 1)m) =n∑

k=0

sxm

n,k(xm)k, (22)

4

(xm)n =n∑

k=0

Sxm

n,k

k∏

j=1

(xm − (j − 1)m), and (23)

∑

n≥0

Sxm

n,kxn =xk

(1− 1mx)(1− 2mx) · · · (1− kmx). (24)

Further Study: Are there exponential and logarithmic generating functions for xm-Stirling numbers?Do numbers which are strongly related to Stirling numbers, such as Bell, Bernoulli, and Eulernumbers, have similar analogues?

Q-Analogues of Poly-Stirling Numbers.

There are also two natural q-analogues of Poly-Stirling numbers of the first and second kind,listed below, both of which have combinatorial interpretations:

Sp(x)n+1,k(q) = S

p(x)n,k−1(q) + p([k]q)S

p(x)n,k (q), (25)

sp(x)n+1,k(q) = s

p(x)n,k−1(q)− p([n]q)s

p(x)n,k (q), (26)

Sp(x)

n+1,k(q) = Sp(x)

n,k−1(q) + [p(k)]qSp(x)

n,k (q), and (27)

sp(x)n+1,k(q) = s

p(x)n,k−1(q)− [p(n)]qs

p(x)n,k (q). (28)

References

[1] K. S. Briggs and J. B. Remmel, A p, q-analogue of a Formula of Frobenius, ElectronicJournal of Combinatorics 10(1) (2003), #R9.

[2] R. Ehrenborg, J. Haglund, and M. A. Readdy, Colored Juggling Patterns and WeightedRook Placements, Unpublished Manuscript.

[3] A. M. Garsia and J. B. Remmel, Q-Counting Rook Configurations and a Formula of Frobe-nius, J. Combin. Theory Ser. A 41 (1986), 246-275.

[4] I. M. Gessel, Generalized Rook Polynomials and Orthogonal Polynomials, q-Series and Par-titions, The IMA Volumes in Mathematics and Its Applications, 1989

[5] J. Goldman and J. Haglund Generalized Rook Polynomials, J. Combin. Theory Ser. A 91(2000), 509-530.

[6] J. R. Goldman, J. T. Joichi, and D. E. White, Rook Theory I. Rook equivalence of Ferrersboards, Proc. Amer. Math Soc. 52 (1975), 485-492.

[7] H. G. Gould, The q-Stirling numbers of the first and second kinds, Duke Math. J. 28 (1961),281-289.

[8] J. Haglund and J. B. Remmel, Rook Theory for Perfect Matchings, Advances in AppliedMathematics 27 (2001), 438-481.

5

[9] L. C. Hsu and P. J. S. Shiue, A Unified Approach to Generalized Stirling Numbers, Advancesin Applied Mathematics 20 (1998), 366-384.

[10] I. Kaplansky and J. Riordan, The problem of rooks and its applications, Duke Math. J. 13(1946), 259-268.

[11] A. de Medicis and P. Leroux, Generalized Stirling Numbers, Convolution Formulae andp, q-analogues, Can. J. Math. 47 (1995), 474-499.

[12] S. C. Milne, Inversion Properties of Triangular Arrays of Numbers, Analysis 1 (1981), 1-7.

[13] J. B. Remmel and M. Wachs, Generalized p, q-Stirling numbers, private communication.

[14] J. Riordan, Combinatorial Identities, Wiley, New York, 1968.

[15] R. Stanley, “Enumerative Combinatorics: Volume II,” Cambridge, 1999

[16] M. Wachs, σ-Resticted growth functions and p, q-Stirling numbers, J. Combin. Theory Ser. A68 (1994), 470-480.

[17] M. Wachs and D. White, p, q-Stirling numbers and set partition statistics, J. Combin. TheorySer. A 56 (1991), 27-46.

6