scholarly journals Elliptic rook and file numbers

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Michael J. Schlosser ◽  
Meesue Yoo
Keyword(s):  
Stirling Numbers ◽  
Classical Case ◽  
Factorization Theorem ◽  
Rook Numbers ◽  
International Audience ◽  
Number Case

International audience In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.

scholarly journals Elliptic Rook and File Numbers

10.37236/6121 ◽  
2017 ◽  
Vol 24 (1) ◽  
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.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

scholarly journals Negative $q$-Stirling numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Yue Cai ◽  
Margaret Readdy
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Argument ◽  
International Audience

International audience The notion of the negative $q$-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative $q$-binomial, we show the classical $q$ -Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $(1+q)$. We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s $q=-1$ phenomenon. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting $t=1+q$ we give a bijective combinatorial argument à la Viennot showing the $(q; t)$-Stirling numbers of the first and second kind are orthogonal. La notion de la $q$-binomial négative était introduite par Fu, Reiner, Stanton et Thiem. Réfléchissant la $q$-binomial négative, nous démontrons que les classiques $q$-nombres de Stirling de deuxième espèce peuvent être exprimés comme une paire de statistiques sur un sous-ensemble des mots de croissance restreinte. Les expressions résultantes sont les polynômes en $q$ et $1+q$. Nous étendons ce résultat énumératif via une décomposition du poset de Stirling, ainsi que d’une version homologique du $q=-1$ phénomène de Stembridge. Un parallèle énumératif, poset théorique et étude homologique des $q$-nombres de Stirling de première espèce se fait en commençant par la formulation du placement des tours par suite des auteurs de Médicis et Leroux. On laisse $t=1+q$ et on donne les arguments combinatoires et bijectifs à la Viennot qui démontrent que les $(q;t)$-nombres de Stirling de première et deuxième espèces sont orthogonaux.

scholarly journals Tiling Periodicity

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Juhani Karhumaki ◽  
Yury Lifshits ◽  
Wojciech Rytter
Keyword(s):  
Open Problem ◽  
Classical Case ◽  
Side Product ◽  
Compact Representation ◽  
Minimal Size ◽  
Partial Word ◽  
International Audience ◽  
New Type

International audience We contribute to combinatorics and algorithmics of words by introducing new types of periodicities in words. A tiling period of a word w is partial word u such that w can be decomposed into several disjoint parallel copies of u, e.g. a lozenge b is a tiling period of a a b b. We investigate properties of tiling periodicities and design an algorithm working in O(n log (n) log log (n)) time which finds a tiling period of minimal size, the number of such minimal periods and their compact representation. The combinatorics of tiling periods differs significantly from that for classical full periods, for example unlike the classical case the same word can have many different primitive tiling periods. We consider also a related new type of periods called in the paper multi-periods. As a side product of the paper we solve an open problem posted by T. Harju (2003).

scholarly journals Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Stirling Numbers ◽  
Algebraic Methods ◽  
Combinatorial Proof ◽  
Nous Obtenons ◽  
Long Cycle ◽  
Bijective Proof ◽  
International Audience

International audience We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account. Nous étudions le nombre de permutations $\beta$ de $[N]$ avec $m$ cycles telles que $(1 2 \ldots N) \beta^{-1}$ a un seul cycle. Ces nombres apparaissent en tant que coefficients des monômes linéaires des polynômes de Kerov et de Stanley. À l'aide de méthodes algébriques, D. Zagier a trouvé une connexion inattendue avec les nombres de Stirling de taille $N+1$. Nous présentons ici la première preuve combinatoire de son résultat, en introduisant une nouvelle bijection entre des cartes partitionnées et des arbres épineux. De plus, nous obtenons un résultat plus fin, prenant en compte le type des permutations.

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.

scholarly journals A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Miguel Méndez ◽  
Adolfo Rodríguez
Keyword(s):  
Stirling Numbers ◽  
Direct Application ◽  
Bell Numbers ◽  
Bijective Correspondence ◽  
Rook Placements ◽  
Combinatorial Model ◽  
International Audience ◽  
Generalized Stirling Numbers ◽  
Dual Graded Graphs ◽  
Combinatorial Methods

International audience We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.

scholarly journals Bijections on m-level Rook Placements

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Kenneth Barrese ◽  
Bruce Sagan
Keyword(s):  
Special Class ◽  
Symmetric Functions ◽  
Elementary Symmetric Functions ◽  
Rook Placements ◽  
Rook Numbers ◽  
Ferrers Board ◽  
International Audience ◽  
Ferrers Boards

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.

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

10.37236/1702 ◽  
2003 ◽  
Vol 10 (1) ◽  
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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