Gessel polynomials, rooks, and extended Linial arrangements

Tewari, Vasu

doi:10.1016/j.jcta.2018.11.017

Cited by 6 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we shall state a bijection between rook placements on 2δ n and increasing binary trees on n + 1 vertices. Our bijection is rather simple, and is, in fact, a special case of a bijection constructed by Tewari in [14]. Tewari's bijection is between maximal rook placements on the board (n − 1) × kn and labelled rooted plane k-ary trees.…”

Section: Introductionmentioning

confidence: 95%

Bijection Between Increasing Binary Trees and Rook Placements on Double Staircases

Deb

2023

Electron. J. Combin.

View full text Add to dashboard Cite

In this paper, we shall construct a bijection between rook placements on double staircases (introduced by Josuat-Vergès in 2017) and increasing binary trees. We introduce two subclasses of rook placements on double staircases, which we call left and right-aligned rook placements. We show that their enumeration, while keeping track of a certain statistic, gives the $\gamma$-vectors of the Eulerian polynomials. We conclude with a discussion on a different bijection that fits in very well with our main bijection, and another discussion on generalising our main bijection. Our main bijection is a special case of a bijection due to Tewari (2019).

show abstract

Section: Introductionmentioning

confidence: 95%

Bijection Between Increasing Binary Trees and Rook Placements on Double Staircases

Deb

2023

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…Note that the ascent-descent statistics on labeled binary trees were first studied in unpublished work by Gessel, and functional equations for their distribution were established by Kalikow [28] and Drake [12]. Given the striking observations of Gessel [16] relating the distribution of these statistics to enumerative questions in the theory of hyperplane arrangements, much work has been done recently [7,11,14,17,44]. That said, we will not focus on the hyperplane arrangements perspective in this article.…”

Section: Introductionmentioning

confidence: 99%

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Tewari

Willigenburg

2019

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations (σ 1 , σ 2 ) that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a 0-Hecke action on permuted composition tableaux. This action naturally defines an equivalence relation on these tableaux. Certain distinguished representatives of the resulting equivalence classes in the special case of two-columned tableaux are in bijection with compatible pairs. We provide a bijection between two-columned tableaux and labeled binary trees. This bijection maps a quadruple of descent statistics for 2-columned tableaux to left and right ascent-descent statistics on labeled binary trees introduced by Gessel, and we use it to prove that the number of compatible pairs is (n + 1) n−1 . 14 4.2. Bijection between LD n and SPCT((2 n )) 15 4.3. Plane binary trees, unlabeled and labeled 162010 Mathematics Subject Classification. Primary 05E10, 20C08; Secondary 05A05, 05A19, 05C20, 05E05, 05E15.

show abstract

“…The first author observed that certain evaluations of B n coincide with the number of regions in various well-known deformations of Coxeter arrangements [21]. This viewpoint has been pursued in [10,15,60], and a complete explanation has been offered by Bernardi [7]. Given a subset A of t´1, 0, 1u, we can consider the arrangement in R n consisting of all hyperplanes x i ´xj " a where i ă j and a P A.…”

Section: Introductionmentioning

confidence: 99%

Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

Gessel

Tewari

2019

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

In 1995, the first author introduced a multivariate generating function G that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that G is symmetric, he conjectured that G is Schur-positive. We prove this conjecture by expanding G positively in terms of ribbon Schur functions. We obtain this expansion using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture showing that the generating function restricted to labeled binary trees with a fixed canopy is still Schur-positive.We also discuss applications in the setting of hyperplane arrangements. We show that a certain specialization of G equals the Frobenius characteristic of the natural S n -action on regions of the semiorder arrangement, which we then expand in terms of the Frobenius characteristics of Foulkes characters. We also construct an S n -action on regions of the Linial arrangement using a set of trees studied by Bernardi, and subsequently compute the character of this action by employing Lagrange inversion. The resulting expression generalizes Postnikov's formula for the number of regions in the Linial arrangement. As a final application, we prove γ-nonnegativity for the distribution of the number of right edges over local binary search trees.

show abstract

Gessel polynomials, rooks, and extended Linial arrangements

Cited by 6 publications

References 29 publications

Bijection Between Increasing Binary Trees and Rook Placements on Double Staircases

Bijection Between Increasing Binary Trees and Rook Placements on Double Staircases

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

Contact Info

Product

Resources

About