The rook numbers of Ferrers boards and the related restricted permutation numbers

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.

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Elliptic Rook and File Numbers

The Electronic Journal of Combinatorics ◽

10.37236/6121 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 3

Author(s):

Michael J. Schlosser ◽

Meesue Yoo

Keyword(s):

Stirling Numbers ◽

Classical Case ◽

Factorization Theorem ◽

Rook Numbers ◽

Generalized Stirling Numbers ◽

Independent Parameters ◽

Ferrers Boards

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 $p$. The elliptic rook numbers are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and extended to the $q$-case by Garsia and Remmel. We obtain similar results for elliptic analogues of Garsia and Remmel's $q$-file numbers for skyline boards. We also provide an elliptic extension of the $j$-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and $r$-restricted versions thereof.

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A $p,q$-analogue of a Formula of Frobenius

The Electronic Journal of Combinatorics ◽

10.37236/1702 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 5

Author(s):

Karen S. Briggs ◽

Jeffrey B. Remmel

Keyword(s):

Stirling Numbers ◽

Combinatorial Interpretation ◽

Eulerian Polynomials ◽

Major Index ◽

Rook Placements ◽

Rook Numbers ◽

Nonnegative Coefficients ◽

Ferrers Boards

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the $q$-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized $p,q$-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a $p,q$-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach's definition and an extension of Garsia and Remmel's proof to give a combinatorial interpretation to a $p,q$-analogue of a formula of Frobenius relating the $p,q$-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over $S_n$. We further define a $p,q$-analogue of the hit numbers, and show analytically that for Ferrers boards, the $p,q$-hit numbers are polynomials in $(p,q)$ with nonnegative coefficients.

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On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

The Electronic Journal of Combinatorics ◽

10.37236/6699 ◽

2017 ◽

Vol 24 (2) ◽

Cited By ~ 2

Author(s):

Sen-Peng Eu ◽

Tung-Shan Fu ◽

Yu-Chang Liang ◽

Tsai-Lien Wong

Keyword(s):

Weyl Algebra ◽

Stirling Numbers ◽

Stirling Number ◽

Combinatorial Structures ◽

Normal Ordering ◽

Rook Placements ◽

Threshold Graphs ◽

Combinatorial Interpretations ◽

Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

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Permutation endomorphisms and refinement of a theorem of Birkhoff

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100034629 ◽

1960 ◽

Vol 56 (4) ◽

pp. 322-328 ◽

Cited By ~ 12

Author(s):

H. K. Farahat ◽

L. Mirsky

Keyword(s):

Abelian Group ◽

Finite Number ◽

Linear Combination ◽

Finite Linear Combination ◽

Usual Manner ◽

Restricted Permutation ◽

Image Position ◽

Ring Of Endomorphisms ◽

Finite Linear

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

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