Combinatorics of generalized q-Euler numbers

Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two q-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain q-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of the same shape. One conjecture is a determinantal formula for the number of pleasant diagrams in terms of Schröder paths and the other conjecture is a determinantal formula for the generating function for RPPs of a skew staircase shape in terms of q-Euler numbers.In this paper, we show that the results of Morales, Pak and Panova on the q-Euler numbers can be derived from previously known results due to Prodinger by manipulating continued fractions. These q-Euler numbers are naturally expressed as generating functions for alternating permutations with certain statistics involving maj. It has been proved by Huber and Yee that these q-Euler numbers are generating functions for alternating permutations with certain statistics involving inv. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving maj to the statistic involving inv. We also prove the aforementioned two conjectures of Morales, Pak and Panova.

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“…There are similar q-Euler numbers studied in [8,18]. We show that these q-Euler numbers have similar properties, see Theorems 4.2 and 4.3.…”

supporting

confidence: 71%

“…The q-Euler numbers E * n (q) have also been studied by Huber and Yee [8] who showed that (9) E * 2n+1 (q) = π∈Alt2n+1 q inv(π)−ndes(πe) , where ndes(π e ) is the number of non-descents in π e = π 2 π 4 · · · π n/2 . Again, by (4) and (9), we have (10) π∈Alt2n+1 q maj(κ2n+1π −1 ) = π∈Alt2n+1 q inv(π)−ndes(πe) .…”

mentioning

confidence: 99%

Reverse plane partitions of skew staircase shapes and q-Euler numbers

Hwang

Kim

Yoo

et al. 2019

Journal of Combinatorial Theory, Series A

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“…Ismail and Zhang [18] conjectured that the coefficients in these polynomials are symmetric about the middle coefficient. This symmetry is explained in [17] from the arithmetic interpretation given in Theorem 3.5.…”

Section: )mentioning

confidence: 98%

“…Huber and A.J. Yee supply a direct arithmetic interpretation for the polynomials T 2n+1 (q) in [17]. In addition, these authors undertake a combinatorial analysis of the polynomials appearing in [23] with similar arithmetic interpretations.…”

Section: )mentioning

confidence: 99%

Zeros of generalized Rogers–Ramanujan series: Asymptotic and combinatorial properties

Huber¹

2010

Journal of Approximation Theory

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In this paper we study the properties of coefficients appearing in the series expansions for zeros of generalized Rogers-Ramanujan series. Our primary purpose is to address several conjectures made by M.E.H. Ismail and C. Zhang. We prove that the coefficients in the series expansion of each zero approach rational multiples of π and π 2 as q → 1 − . We also show that certain polynomials arising in connection with the zeros of Rogers-Ramanujan series generalize the coefficients appearing in the Taylor expansion of the tangent function. These polynomials provide an enumeration for alternating permutations different from that given by the classical q-tangent numbers. We conclude the paper with a method for inverting an elliptic integral associated with the zeros of generalized Rogers-Ramanujan series. Our calculations provide an efficient algorithm for the computation of series expansions for zeros of generalized Rogers-Ramanujan series.

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“…The implications and extensions of these results are explored by Huber [27], Huber and Yee [28] and Ismail [29].…”

Section: Surprises In the Lost Notebookmentioning

confidence: 99%