Derangements on a Ferrers board

We study the derangement number on a Ferrers board B = (n × n) - λ with respect to an initial permutation M, that is, the number of permutations on B that share no common points with M. We prove that the derangement number is independent of M if and only if λ is of rectangular shape. We characterize the initial permutations that give the minimal and maximal derangement numbers for a general Ferrers board, and present enumerative results when λ is a rectangle.

Bijections on m-level Rook Placements

International audience Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two 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 Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne. Nous considérons les rangs d’un échiquier partagés en ensembles de $m$ rangs appelés les niveaux. Un $m$-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de $m$-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de $m$-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne.

Cycles and sorting index for matchings and restricted permutations

International audience We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.

Descents of Permutations in a Ferrers Board

The classical Eulerian polynomials are defined by setting $$A_n(t)= \sum_{\sigma \in \mathfrak{S}_n} t^{1+\mathrm{des}(\sigma)}= \sum_{k=1}^n A_{n,k} t^k$$where $A_{n,k}$ is the number of permutations of length $n$ with $k-1$ descents. Let $A_n(t, q) = \sum_{\pi \in \mathfrak{S}_n} t^{1+{\rm des}(\pi)}q^{{\rm inv}(\pi)} $ be the $\mathrm{inv}$ $q$-analogue of the classical Eulerian polynomials whose generating function is well known: \begin{eqnarray}\sum_{n \geq 0} \frac{u^n A_n(t, q)}{[n]_q!} = \frac{1}{\displaystyle 1-t\sum_{k \geq 1} \frac{(1-t)^ku^k}{[k]_q!}}.\qquad\qquad(*)\label{perm_gf abs}\end{eqnarray}In this paper we consider permutations restricted in a Ferrers board and study their descent polynomials. For a general Ferrers board $F$, we derive a formula in the form of permanent for the restricted $q$-Eulerian polynomial $$A_F(t,q) := \sum_{\sigma \in F} t^{1+{\rm des}(\sigma)} q^{{\rm inv}(\sigma)}.$$ If the Ferrers board has the special shape of an $n\times n$ square with a triangular board of size $s$ removed, we prove that $A_F(t,q)$ is a sum of $s+1$ terms, each satisfying an equation that is similar to (*). Then we apply our results to permutations with bounded drop (or excedance) size, for which the descent polynomial was first studied by Chung et al. (European J. Combin., 31(7) (2010):1853-1867). Our method presents an alternative approach.

Augmented Rook Boards and General Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

Classification of Ding's Schubert Varieties: Finer Rook Equivalence

K. Ding studied a class of Schubert varieties Xƛ in type A partial flag manifolds, indexed by integer partitions ƛ and in bijection with dominant permutations. He observed that the Schubert cell structure of Xƛ is indexed by maximal rook placements on the Ferrers board Bƛ, and that the integral cohomology groups H*(Xƛ; ℤ), H*(Xμ; ℤ) are additively isomorphic exactly when the Ferrers boards Bƛ, Bμ satisfy the combinatorial condition of rook-equivalence.We classify the varieties Xƛ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

ROOK PLACEMENTS AND CLASSIFICATION OF PARTITION VARIETIES B\ Mλ

Let M be the set of m by n complex matrices of rank m. Let λ = (λ1,…,λm) be a partition with λi ≥ λi + 1 for 1 ≤ i ≤ m - 1 and λ1 = n. A Ferrers board Fλ is a right justified subarray in a m by n matrix with the length of the ith row being λi, and define Mλ={a ∈ M|ai,j = 0 if (i,j) ∉ Fλ}. Let B be the Borel subgroup of the general linear group GLm (C) consisting of upper triangular matrices. Define B\Mλ={Ba|a∈ Mλ}. The quotient space B\Mλ is a projective variety called a partition variety associated to λ. In this note, we classify partition varieties B\Mλ according to their homology and cohomology groups (up to isomorphisms).

A random walk on the rook placements on a Ferrer's board

Let $B$ be a Ferrers board, i.e., the board obtained by removing the Ferrers diagram of a partition from the top right corner of an $n\times n$ chessboard. We consider a Markov chain on the set $R$ of rook placements on $B$ in which you can move from one placement to any other legal placement obtained by switching the columns in which two rooks sit. We give sharp estimates for the rate of convergence of this Markov chain using spectral methods. As part of this analysis we give a complete combinatorial description of the eigenvalues of the transition matrix for this chain. We show that two extremes cases of this Markov chain correspond to random walks on groups which are analyzed in the literature. Our estimates for rates of convergence interpolate between those two results.