Rook Theory. I.: Rook Equivalence of Ferrers Boards

1975 ◽  
Vol 52 (1) ◽  
pp. 485 ◽  
Author(s):  
Jay R. Goldman ◽  
J. T. Joichi ◽  
Dennis E. White
Keyword(s):  
Rook Theory ◽  
Ferrers Boards ◽  
Rook Equivalence

scholarly journals Rook theory. I. Rook equivalence of Ferrers boards

1975 ◽  
Vol 52 (1) ◽  
pp. 485-485
Author(s):  
Jay R. Goldman ◽  
J. T. Joichi ◽  
Dennis E. White
Keyword(s):  
Rook Theory ◽  
Ferrers Boards ◽  
Rook Equivalence

scholarly journals A Graph Theory of Rook Placements

10.37236/8435 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Kenneth Barrese
Keyword(s):  
Graph Theory ◽  
Equivalence Class ◽  
Connected Graph ◽  
The Other ◽  
Connected Graphs ◽  
Rook Placements ◽  
Ferrers Boards ◽  
Rook Equivalence

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 on an equivalence class 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 single other column. Given such a graph, we characterize 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 definition to the m-level rook placement generalization developed by Briggs and Remmel. This yields a graph on the set of rook equivalent singleton boards, and we characterize which singleton boards give rise to a connected graph.

scholarly journals On criteria for rook equivalence of Ferrers boards

2019 ◽  
Vol 76 ◽  
pp. 199-207 ◽  
Author(s):  
Jonathan Bloom ◽  
Dan Saracino
Keyword(s):  
Ferrers Boards ◽  
Rook Equivalence

scholarly journals On Garsia-Remmel problem of rook equivalence

1996 ◽  
Vol 149 (1-3) ◽  
pp. 59-68 ◽  
Author(s):  
Kequan Ding ◽  
Paul Terwilliger
Keyword(s):  
Rook Equivalence

scholarly journals Augmented Rook Boards and General Product Formulas

10.37236/809 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Brian K. Miceli ◽  
Jeffrey Remmel
Keyword(s):  
Product Formula ◽  
Combinatorial Proof ◽  
Rook Theory ◽  
Special Cases ◽  
Ferrers Board ◽  
General Product ◽  
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.

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

10.37236/6699 ◽  
2017 ◽  
Vol 24 (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.

Rook Theory. II: Boards of Binomial Type

1976 ◽  
Vol 31 (4) ◽  
pp. 618-633 ◽  
Author(s):  
Jay R. Goldman ◽  
J. T. Joichi ◽  
David L. Reiner ◽  
Dennis E. White
Keyword(s):  
Rook Theory ◽  
Binomial Type

Rooks on ferrers boards and matrix integrals

1999 ◽  
Vol 96 (5) ◽  
pp. 3531-3536 ◽  
Author(s):  
S. V. Kerov
Keyword(s):  
Matrix Integrals ◽  
Ferrers Boards

scholarly journals Some combinatorial identities involving noncommuting variables

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Michael Schlosser ◽  
Meesue Yoo
Keyword(s):  
Functional Equations ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Weight Functions ◽  
Nous Proposons ◽  
Nous Obtenons ◽  
Exponential Functions ◽  
Rook Theory ◽  
International Audience ◽  
Nous Utilisons

International audience We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers. Nous obtenons des identités combinatoires pour des variables satisfaisant des ensembles spécifiques de relations de commutation. Ces identités ainsi obtenues généralisent leurs analogues pour des variables $q$-commutantes $x$ et $y$ satisfaisant $yx=qxy$. En particulier, nous obtenons des théorèmes binomiaux dépendant du poids, des équations fonctionnelles pour les fonctions exponentielles généralisées, nous proposons une dérivée des variables non-commutatives, et finalement nous utilisons l’une des fonctions de poids considérées pour étendre la théorie des tours. Nous en déduisons une généralisation des $q$-nombres de Stirling de seconde espèce et des $q$-nombres de Lah.

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