Enumeration of permutations by number of alternating descents

Ma, Shi-Mei; Yeh, Yeong‐Nan

doi:10.1016/j.disc.2015.12.007

Cited by 14 publications

(14 citation statements)

References 13 publications

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“…Interestingly, Q 1 (v) = v generates the same sequence of polynomials for n 2. 5 12 n The generating polynomials for the numbers of alternating descents (π(i) ≷ π(i+1) depending on the parity of i) or for the number of 3-descents (either of the patterns 132, 213 or 321) satisfy (see [47,164])…”

Section: Polynomials With Quadratic α(V)mentioning

confidence: 99%

“…A very interesting property of P n (v) is that all roots lie on the left half unit circle, namely, v = e iθ with 1 2 π θ 3 2 π; see [164] for more information and Figure 4 for an illustration. Such a root-unitary property implies an alternative proof of the CLT via the fourth moment theorem of [130]: the fourth centered and normalized moment tends to three iff the coefficients are asymptotically normally distributed.…”

Section: Polynomials With Quadratic α(V)mentioning

confidence: 99%

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An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

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We study linear recurrences of Eulerian type of the formwith P 0 (v) given, where α(v), β(v) and γ(v) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of P n (v) for large n using the method of moments and analytic combinatorial tools under varying α(v), β(v) and γ(v), and apply our results to more than two hundred of concrete examples when β(v) = 0 and more than three hundred when β(v) = 0 that we collected from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.

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Section: Polynomials With Quadratic α(V)mentioning

confidence: 99%

Section: Polynomials With Quadratic α(V)mentioning

confidence: 99%

An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

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“…where g i = 2(i + 1)q and h i = (1 + q 2 )i(i + 1) for i ≥ 0. On the other hand, Ma and Yeh [31] recently proved…”

Section: -Q-log-convexity Of Alternating Eulerian Polynomialsmentioning

confidence: 99%

Positivity of Iterated Sequences of Polynomials

Zhu¹

2018

SIAM J. Discrete Math.

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In this paper, we present some criteria for the 2-q-log-convexity and 3-q-logconvexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array [A n,k (q)] n,k≥0 of polynomials in q with nonnegative coefficients satisfying the recurrence relationThose criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the 2-q-log-convexity of alternating Eulerian polynomials, 2-log-convexity of Euler numbers, and 3-q-log-convexity of many classical polynomials, including the Bell polynomials, the Eulerian polynomials of Types A and B, the q-Schröder numbers, q-central Delannoy numbers, the Narayana polynomials of Types A and B, the generating functions of rows in the Catalan triangles of Aigner and Shapiro, the generating functions of rows in the large Schröder triangle, and so on, which extend many known results for q-log-convexity.MSC: 05A20; 05A30; 11B37; 11B83; 30B70

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“…Moreover, Gessel and Zhuang [36] extended some results in [27,63] by using noncommutative symmetric functions. For n ≥ 1, Ma and Yeh [57] gave the explicit formula and the recurrence relation…”

Section: Alternating Descents Of Permutationsmentioning

confidence: 99%

“…Remark 4.15. We refer the reader to [47,57,81] for the corresponding different proof for Theorem 4.14. Integrating with respect to (4.21) in t, we recover the exponential generating function of A n (x) occurred in [27,Theorem 4.2] as follows:…”

Section: Alternating Descents Of Permutationsmentioning

confidence: 99%

Stability of combinatorial polynomials and its applications

Ding,

Zhu

2021

Preprint

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Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to make a systematical study on the stability of polynomials in combinatorics.Applying the characterizations of Borcea and Brändén concerning linear operators preserving stability, we present criteria for real stability and Hurwitz stability of recursive polynomials. We also give a criterion for Hurwitz stability of the Turán

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Enumeration of permutations by number of alternating descents

Abstract: In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these numbers.

Cited by 14 publications

References 13 publications

An asymptotic distribution theory for Eulerian recurrences with applications

An asymptotic distribution theory for Eulerian recurrences with applications

Positivity of Iterated Sequences of Polynomials

Stability of combinatorial polynomials and its applications

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