A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have identical Schubert cell structures. This also produces a complete classification of isomorphism types of lower intervals of 312-avoiding permutations in the Bruhat order.
We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations pi such that [id,pi] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincare polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincare polynomial of some particularly interesting intervals in the finite Weyl grou…
In [2] we introduced a new notion of Wilf equivalence of integer partitions and proved that rook equivalence implies Wilf equivalence. In the present paper we prove the converse and thereby establish a new criterion for rook equivalence. We also refine two of the standard criteria for rook equivalence and establish another new one involving what we call \emph{nested sequences of L's}.
Rook polynomials have been studied extensively since 1946, principally as a method for enumerating restricted permutations. However, they have also been shown to have many fruitful connections with other areas of mathematics, including graph theory, hypergeometric series, and algebraic geometry. It is known that the rook polynomial of any board can be computed recursively. The naturally arising inverse question -- given a polynomial, what board (if any) is associated with it? -- remains open. In this paper, we solve the inverse problem completely for the class of Ferrers boards, and show that …
Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of $m$ rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We generalize this result to any Ferrers board as well as giving a p,q-analogue. We also consider a dual situation involving weighted file placements which permit more than one rook in the same row. In b…
Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes a map by Foata and Sch\"utzenberger and our proof applies to any Ferrers board. This bijection also preserves the m-inversion number statistic of an m-level rook placement, defined by Briggs and Remmel…
Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph of an equivalence set of Ferrers boards by specifying that two boards are connected by an edge if you can obtain one of the boards by moving squares in the other board out of one column and into a singe other column. Given such a graph, we categorize which boards will yield connected graphs. We also provide some cases where common graphs will or will not be the graph for some set of rook equivalent Ferrers boards. Finally, we extend this graph de…
K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups H^*(X_λ; Zz), H^*(X_μ; Zz) are additively isomorphic exactly when the Ferrers boards B_λ, B_μsatisfy the combinatorial condition of rook-equivalence. We classify the varieties X_λup to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. Th…
We study the simplicial complex that arises from non-attacking rook placements on a subclass of Ferrers boards that have $a_i$ rows of length $i$ where $a_i>0$ and $i\leq n$ for some positive integer $n$. In particular, we will investigate enumerative properties of their facets, their homotopy type, and homology.
In 2009, Etzion and Siberstein proposed a conjecture on the largest dimension of a linear space of matrices over a finite field in which all nonzero matrices are supported on a Ferrers diagram and have rank bounded below by a given integer. Although several cases of the conjecture have been established in the past decade, proving or disproving it remains to date a wide open problem. In this paper, we take a new look at the Etzion-Siberstein Conjecture, investigating its connection with rook theory. Our results show that the combinatorics behind this open problem is closely linked to the theory…
Connections between $q$-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel's $q$-hit polynomial. Both this new statistic $mat$ and another statistic for the $q$-hit polynomial $ξ$ recently introduced by Dworkin are shown to induce different multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are shown to generate different families of Euler-Mahonian statistics. For these boards the $ξ$ family includes Denert's statistic $den$, and gives a new proof of Foata and Zeilberger's Theor…
We derive a combinatorial equilibrium for bounded juggling patterns with a random, $q$-geometric throw distribution. The dynamics are analyzed via rook placements on staircase Ferrers boards, which leads to a steady-state distribution containing $q$-rook polynomial coefficients and $q$-Stirling numbers of the second kind. We show that the equilibrium probabilities of the bounded model can be uniformly approximated with the equilibrium probabilities of a corresponding unbounded model. This observation leads to new limit formulae for $q$-analogues. Keywords: juggling pattern; $q$-Stirling number…
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-…
We investigate the rank-generating function $F_λ$ of the poset of partitions contained inside a given shifted Ferrers shape $λ$. When $λ$ has four parts, we show that $F_λ$ is unimodal when $λ=\langle n,n-1,n-2,n-3 \rangle$, for any $n\ge 4$, and that unimodality fails for the doubly-indexed, infinite family of partitions of the form $λ=\langle n,n-t,n-2t,n-3t \rangle$, for any given $t\ge 2$ and $n$ large enough with respect to $t$. When $λ$ has $b\le 3$ parts, we show that our rank-generating functions $F_λ$ are all unimodal. However, the situation remains mostly obscure for $b\ge 5$. In gen…
In a recent paper, Backelin, West and Xin describe a map $ϕ^*$ that recursively replaces all occurrences of the pattern $k... 21$ in a permutation $σ$ by occurrences of the pattern $(k-1)... 21 k$. The resulting permutation $ϕ^*(σ)$ contains no decreasing subsequence of length $k$. We prove that, rather unexpectedly, the map $ϕ^*$ commutes with taking the inverse of a permutation. In the BWX paper, the definition of $ϕ^*$ is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map $ϕ^*$ is the key step in provi…
