A combinatorial proof of a tantalizing symmetry on Catalan objects

Bóna, Miklós; Dimitrov, Stoyan; Labelle, Gilbert; Li, Yifei; Pappe, Joseph; Vindas-Meléndez, Andrés R.; Zhou, Yan

doi:10.48550/arxiv.2212.10586

Cited by 1 publication

(8 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Case B1: n = 2k − 1. In this case, we have w n,k,m−1 = 0 by (1). On the other hand, the second term on the right-hand side of (4) naturally vanishes.…”

Section: Degmentioning

confidence: 96%

“…Case A2: n = k. In this case, the third term on the right-hand side of (4) naturally vanishes. Note that for m = 1 we have w n+1,k,m−1 = 0 by (1). Thus, it suffices to show that…”

Section: Degmentioning

confidence: 96%

“…This paper is much motivated by a recent work [1] To obtain this result, they derived the following explicit formula for the numbers w n,k,m by using generating function techniques.…”

Section: Introductionmentioning

confidence: 99%

“…Let W n,k (x) be the generating polynomial of w n,k,m as given by The real-rootedness of W n,k (x) was proved by Bóna et al [1] based on Malos result regarding the roots of the Hadamard product of two real-rooted polynomials. It is worth mentioning that many combinatorial polynomials have only real zeros.…”

Section: Introductionmentioning

confidence: 99%

“…Bóna et al [1] further studied the interlacing property of W n,k (x) by fixing n or k, and proposed two interesting conjectures. Before stating these conjectures, let us recall some related definitions following Liu and Wang [8].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Interlacing Property of a Family of Generating Polynomials over Dyck Paths

Wang,

Zhang

2024

Electron. J. Combin.

View full text Add to dashboard Cite

In the study of a tantalizing symmetry on Catalan objects, Bóna et al. introduced a family of polynomials $\{W_{n,k}(x)\}_{n\geq k\geq 0}$ defined by$$W_{n,k}(x)=\sum_{m=0}^{k}w_{n,k,m}x^{m},$$where $w_{n,k,m}$ counts the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. They proposed two conjectures on the interlacing property of these polynomials, one of which states that $\{W_{n,k}(x)\}_{n\geq k}$ is a Sturm sequence for any fixed $k\geq 1$, and the other states that $\{W_{n,k}(x)\}_{1\leq k\leq n}$ is a Sturm-unimodal sequence for any fixed $n\geq 1$. In this paper, we obtain certain recurrence relations for $W_{n,k}(x)$, and further confirm their conjectures.

show abstract

“…Case B1: n = 2k − 1. In this case, we have w n,k,m−1 = 0 by (1). On the other hand, the second term on the right-hand side of (4) naturally vanishes.…”

Section: Degmentioning

confidence: 96%

“…Case A2: n = k. In this case, the third term on the right-hand side of (4) naturally vanishes. Note that for m = 1 we have w n+1,k,m−1 = 0 by (1). Thus, it suffices to show that…”

Section: Degmentioning

confidence: 96%

“…This paper is much motivated by a recent work [1] To obtain this result, they derived the following explicit formula for the numbers w n,k,m by using generating function techniques.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations