scholarly journals Combinatorial Interpretations of the Kreweras Triangle in Terms of Subset Tuples

10.37236/7531 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Combinatorial Interpretations ◽  
Dumont Permutations

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

scholarly journals Combinatorial Study of Dellac Configurations and $q$-Extended Normalized Median Genocchi Numbers

10.37236/4068 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Flag Varieties ◽  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Degenerate Flag Varieties

In two recent papers, Feigin proved that the Poincaré polynomials of the degenerate flag varieties have a combinatorial interpretation through Dellac configurations, and related them to the $q$-extended normalized median Genocchi numbers $\bar{c}_n(q)$ introduced by Han and Zeng, mainly by geometric considerations. In this paper, we give combinatorial proofs of these results by constructing statistic-preserving bijections between Dellac configurations and two other combinatorial models of $\bar{c}_n(q)$.

scholarly journals A new approach to the r-Whitney numbers by using combinatorial differential calculus

2019 ◽  
Vol 11 (2) ◽  
pp. 387-418
Author(s):  
Miguel A. Méndez ◽  
José L. Ramírez
Keyword(s):  
Differential Calculus ◽  
Differential Operators ◽  
Family Study ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Combinatorial Interpretation ◽  
New Approach ◽  
Whitney Numbers ◽  
Combinatorial Interpretations ◽  
Binomial Type

Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

scholarly journals Path tableaux and combinatorial interpretations of immanants for class functions on $S_n$

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sam Clearman ◽  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Irreducible Character ◽  
Symmetric Function ◽  
Nonnegative Matrix ◽  
Schur Function ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations ◽  
Characteristic Map ◽  
International Audience ◽  
Totally Nonnegative Matrix

International audience Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$. Soit $χ ^λ$ le caractère irréductible de $S_n$ qui correspond à la partition λ de l'entier n, ou de manière équivalente, la préimage de la fonction de Schur $s_λ$ par l'application caractéristique de Frobenius. Soit $\phi ^λ$ la fonction $S_n →ℂ$ qui est la préimage de la fonction symétrique monomiale m_λ . La valeur du caractère irréductible immanent $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ est non négative pour chaque matrice totalement non négative. Nous donnons une interprétation combinatoire de la valeur $Imm_λ (A)$ lorsque $λ$ est une partition en équerre. Stembridge a conjecturé que la valeur de l'immanent monomial $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ de $\phi ^λ$ est elle aussi non négative pour chaque matrice totalement non négative. Nous confirmons cette conjecture quand λ satisfait $λ _1 ≤2$, et nous donnons une interprétation combinatoire de $Imm_{{\phi} ^λ} (A)$ dans ce cas.

scholarly journals Combinatorics of the Three-Parameter PASEP Partition Function

10.37236/509 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Exclusion Process ◽  
Asymmetric Exclusion Process ◽  
Combinatorial Interpretation ◽  
Rook Placements ◽  
Combinatorial Interpretations ◽  
The Matrix ◽  
The One ◽  
New Formula

We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams. We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generating function of permutations with respect to right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection of Françon and Viennot. Then we give a new formula for the partition function which generalizes the one of Blythe & al. It is proved in two combinatorial ways. The first proof is an enumeration of lattice paths which are known to be a solution of the Matrix Ansatz of Derrida & al. The second proof relies on a previous enumeration of rook placements, which appear in the combinatorial interpretation of a related normal ordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.

scholarly journals Combinatorial Interpretations of the Jacobi-Stirling Numbers

10.37236/342 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yoann Gelineau ◽  
Jiang Zeng
Keyword(s):  
Spectral Theory ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations ◽  
Central Factorial Numbers

The Jacobi-Stirling numbers of the first and second kinds were introduced in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.

scholarly journals The Complete cd-Index of Boolean Lattices

10.37236/5100 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Neil J.Y. Fan ◽  
Liao He
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Boolean Lattice ◽  
Combinatorial Interpretation ◽  
Euler Numbers ◽  
Combinatorial Interpretations ◽  
Bruhat Graph ◽  
Boolean Lattices ◽  
One To One ◽  
Entringer Numbers

Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.

scholarly journals On the Positive Moments of Ranks of Partitions

10.37236/3852 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Kathy Q. Ji ◽  
Erin Y. Y. Shen
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of the $2k$-th symmetrized moment $\eta_{2k}(n)$ of ranks of partitions of $n$ in terms of $(k+1)$-marked Durfee symbols of $n$. In this paper, we consider the $k$-th symmetrized positive moment $\bar{\eta}_k(n)$ of ranks of partitions of $n$ which is defined as the truncated sum over positive ranks of partitions of $n$. As combinatorial interpretations of $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$, we show that for given $k$ and $i$ with $1\leq i\leq k+1$, $\bar{\eta}_{2k-1}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being zero and $\bar{\eta}_{2k}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being positive. The interpretations of $\bar{\eta}_{2k-1}(n)$ and $\bar{\eta}_{2k}(n)$ are independent of $i$, and they imply the interpretation of $\eta_{2k}(n)$ given by Andrews since $\eta_{2k}(n)$ equals $\bar{\eta}_{2k-1}(n)$ plus twice of $\bar{\eta}_{2k}(n)$. Moreover, we obtain the generating functions for $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$.

scholarly journals On q-poly-Bernoulli numbers arising from combinatorial interpretations

2021 ◽  
Author(s):  
Beáta Bényi ◽  
José L. Ramírez
Keyword(s):  
Bernoulli Numbers ◽  
Analytical Results ◽  
Combinatorial Interpretations

AbstractIn this paper we present several natural q-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some related analytical results and ask for combinatorial interpretations.

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