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Touchard type identity for q-narayana numbers

WebSep 30, 2024 · We study the statistics area, bounce and dinv associated to polyominoes in a rectangular box m times n. We show that the bi-statistics (area, bounce) and (area, dinv) … WebThe q −Narayana numbers are polynomials in q with integer coefficients. The first values of these q −Narayana numbers are given in the following table: 1 q1 q3 2q+q2 1 q6 q2 +2q3 …

Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers …

WebVolume 218, Issue 12, 15 February 2012, Pages 6661-6665. Touchard like polynomials and generalized Stirling numbers. Author links open overlay panel G. Dattoli a, B. Germano b, … WebON TOUCHARD’S CONTINUED FRACTION AND EXTENSIONS: COMBINATORICS ... We did not go for an exhaustive list of continued fractions of the Touchard type, in order to keep this paper short and crisp. However, we are ... Our goal will be achieved once we are able to establish the identity B(q) = B(1) X k≥0 (−1)kq(k+1 2)vk. TOUCHARD’S CONTINUED ... piggly wiggly bush la weekly ad https://aladinweb.com

Touchard type identity for q-Narayana numbers - ResearchGate

WebMar 13, 2024 · The behavior of a program that adds specializations for type_identity is undefined. Contents. 1 Member types; 2 Helper types; 3 Possible implementation; 4 Notes; 5 Example Member types. Name Definition type: T Helper types. template < class T > using type_identity_t = typename type_identity < T >:: type; Webdefine the type Bq-Eulerian number EB n,k(q) as follows: EB n,k(q) = X π∈Bn ⌊fwex(π)/2⌋=k qcr(π). Theorem 2.1. For any 0 ≤ k≤ n, we have EB n,k(−1) = n k, EB n,k(0) = n k 2 (the Narayana numbers of type B), En,k(1) is the type BEulerian number, and (4) EB n,k(q) = EB n,n−k(q). In Section 5 we consider crossings and alignments ... WebAbstract. We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur … piggly wiggly broad st sumter

A001263 - OEIS

Category:Rational Catalan Combinatorics: The Associahedron - Inria

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Touchard type identity for q-narayana numbers

The q-log-convexity of the Narayana polynomials of type B

WebOct 10, 2014 · Matthieu Josuat-Verges, A q-analog of Schläfli and Gould identities on Stirling numbers, Preprint, 2016. S. Kamioka, Laurent biorthogonal polynomials, q-Narayana … WebMay 1, 2024 · Abstract. We prove a Touchard type identity for q-Narayana number as follows:1 [n]q [nk]q [nk+1]q=∑h=0min⁡ {k,n−1−k}Ch⋅qh2 (n−1n−1−h−k,2h,k−h)q, where …

Touchard type identity for q-narayana numbers

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WebDe nitions q-Narayana NumbersNarayana polynomialsSome Open Problems q-Narayana Numbers The q-Narayana numbers, as a natural q-analogue of the Narayana numbers N(n;k), arise in the study of q-Catalan numbers. The q-Narayana number N q(n;k)is given by N q(n;k) = 1 [n] hn k i n k 1 qk2 k; where we have adopted the common notation WebNamely, for $0\leq k \leq r$ , the kth W-Narayana number $\operatorname {\mathrm {Nar}} ... (\Phi )/W$ , the number of nonnesting partitions of type X is the same as the number of …

WebMay 1, 2024 · We prove a Touchard type identity for q-Narayana number as follows: 1 [n] q [n k] q [n k + 1] q = ∑ h = 0 min ⁡ {k, n − 1 − k} C h ⋅ q h 2 (n − 1 n − 1 − h − k, 2 h, k − h) q, … WebAbstract. We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of prod-ucts of elementary symmetric functions in terms of Schur functions with nonnegative coecients.

WebWe show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q, t-analogue of Narayana numbers, which was introduced by two of these authors in a recent … Web\begin{abstract} We prove the total positivity of the Narayana triangles of type $A$ and type $B$, and thus affirmatively confirm a conjecture of Chen, Liang and Wang ...

Webtheirs. We also study the combinatorial properties of our q-Kreweras numbers, i.e. positivity, relation with q-Narayana numbers, and cyclic sieving phenomenon. 0. Introduction 1 1. Definitions and notations 4 2. Coincidental Coxeter groups 4 3. q-Kreweras numbers attached to limit symbols 5 4. Type An−1 9 5. Type BCn 10 6. Type H3 19 7. Type ...

WebWe prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain … pineywoods pathologyWebOct 26, 2002 · Some New Observations on Triangular Numbers : There are some triangular numbers which are product of three consecutive numbers. For example 120 is a triangular … piggly wiggly cedar grove wisconsinWebThe numbers in type A Narayana numbers as h-vector The definitions in all types Narayana numbers as h-vector of the associahedron Definition The d-dimensionalassociahedronis a simple polytope with (n + 3)-gontriangulationsas vertices,diagonal flipsas edges. (f V. Reiner and E. Sommers q-Narayana, q-Kreweras for Weyl groups piggly wiggly cedar bluffWebDec 15, 2024 · The construction of these numbers resembles the argument of Reiner and Sommers and these two share similar properties, but our version is slightly different from and combinatorially simpler than theirs. piggly wiggly camden scWeb(sequence A001263 in the OEIS) . Combinatorial interpretations Dyck words. An example of a counting problem whose solution can be given in terms of the Narayana numbers ⁡ (,), is … piggly wiggly byhalia mshttp://math.sun.ac.za/prodinger/pdffiles/touchard-2011.pdf piggly wiggly campbellsport wiWebDe nitions q-Narayana NumbersNarayana polynomialsSome Open Problems q-Narayana Numbers The q-Narayana numbers, as a natural q-analogue of the Narayana numbers … piggly wiggly cedarburg wi weekly ad