scholarly journals Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

10.37236/9780 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Gaspard Ohlmann
Keyword(s):  
Combinatorial Proof ◽  
Combinatorial Approach ◽  
Motzkin Path ◽  
Wilson Polynomials ◽  
Motzkin Paths

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where $d=0$. Their values have already been computed by Kim and Stanton in 2015, however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with $d=0$ only, providing the first fully combinatorial proof for that case.

scholarly journals A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials

10.37236/5350 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Maria Monks Gillespie
Keyword(s):  
Macdonald Polynomials ◽  
Combinatorial Proof ◽  
Symmetry Relation ◽  
Combinatorial Approach ◽  
Combinatorial Formula ◽  
Recursive Structure ◽  
Littlewood Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

scholarly journals A COMBINATORIAL PROOF OF TWO EQUIVALENT IDENTITIES BY FREE 2-MOTZKIN PATHS

Integers ◽  
2014 ◽  
Author(s):  
Alina F. Y. Zhao
Keyword(s):  
Combinatorial Proof ◽  
Motzkin Paths

scholarly journals Minimal and maximal plateau lengths in Motzkin paths

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Helmut Prodinger ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Gumbel Distribution ◽  
Minimal Length ◽  
Analytic Method ◽  
Horizontal Segment ◽  
Motzkin Path ◽  
Dyck Paths ◽  
Substitution Process ◽  
International Audience ◽  
Motzkin Paths

International audience The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters <i> maximal plateau length, horizontal segment </i>and <i>maximal horizontal segment </i>as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

scholarly journals Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns

10.37236/1687 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Yu-Ping Deng ◽  
Laura L. M. Yang
Keyword(s):  
Recent Result ◽  
Motzkin Path ◽  
Reduced Decompositions ◽  
Inversion Number ◽  
Forbidden Patterns ◽  
Motzkin Paths

We obtain a characterization of $(321, 3\bar{1}42)$-avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of $(321,3\bar{1}42)$-avoiding permutations of length $n$ equals the $n$-th Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo, Pergola, Pinzani and Guibert. Similarly, we obtain a characterization of $(231,4\bar{1}32)$-avoiding permutations. For these two classes, we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover, we find a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path.

scholarly journals CONGRUENCES MODULO SQUARES OF PRIMES FOR FU'S k DOTS BRACELET PARTITIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 939-943 ◽  
Author(s):  
CRISTIAN-SILVIU RADU ◽  
JAMES A. SELLERS
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Infinite Family ◽  
Combinatorial Proof ◽  
Combinatorial Approach ◽  
Powers Of 2 ◽  
The Family ◽  
Fixed Positive Integer ◽  
Congruence Properties

In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by 𝔅k(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, [Formula: see text] We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function 𝔅7.

scholarly journals A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Michelle Snider
Keyword(s):  
Combinatorial Proof ◽  
Combinatorial Approach ◽  
Möbius Inversion ◽  
International Audience ◽  
K Theory ◽  
Multiplicity Free ◽  
Grothendieck Polynomials

International audience We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong. On examine la règle de Buch pour la K-théorie de la variété grassmannienne dans les cas sans multiplicité de Schur, qui ont étés classifiés par Stembridge. En utilisant un résultat de Knutson, on démontre que les coefficients de Buch sont liés à l'inversion de Möbius. On en fait une preuve directe et combinatoire qui passe par le developpement de produits de polynômes de Grothendieck. Pour conclure, on donne une application de cette théorie aux cas sans multiplicité de Thomas et Yong.

scholarly journals A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maria Monks Gillespie
Keyword(s):  
Young Diagram ◽  
Macdonald Polynomials ◽  
Point Of View ◽  
Combinatorial Proof ◽  
Symmetry Relation ◽  
Combinatorial Approach ◽  
Combinatorial Formula ◽  
Monomial Basis ◽  
Littlewood Polynomials ◽  
International Audience

International audience We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.

scholarly journals Combinatorial Proofs of Addition Formulas

10.37236/4793 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Xiang-Ke Chang ◽  
Xing-Biao Hu ◽  
Hongchuan Lei ◽  
Yeong-Nan Yeh
Keyword(s):  
Hankel Matrix ◽  
Direct Consequence ◽  
Combinatorial Proof ◽  
Addition Formula ◽  
Polynomial Sequence ◽  
Schröder Paths ◽  
Motzkin Paths ◽  
Addition Formulas

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.

A Combinatorial Approach to Some Sparse Matrix Problems.

1983 ◽  
Author(s):  
S. Thomas McCormick
Keyword(s):  
Sparse Matrix ◽  
Combinatorial Approach ◽  
Matrix Problems
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