On rotated Schur-positive sets

10.37236/8186 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Sheila Sundaram

Keyword(s):

Lexicographic Order ◽

Character Table ◽

Power Sums ◽

Special Cases ◽

In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this conjecture and establish its truth in the following special cases: for $\mu\in [(n-4,1^4), (n)]$ or $\mu\in [(1^n), (3,1^{n-3})], $ or $\mu=(3, 2^k, 1^r)$ when $k\geq 1$ and $0\leq r\leq 2.$ Many new Schur positivity questions are presented.

A Combinatorial Bijection on di-sk Trees

10.37236/10484 ◽

2021 ◽

Vol 28 (4) ◽

Author(s):

Shishuo Fu ◽

Zhicong Lin ◽

Yaling Wang

Keyword(s):

Binary Tree ◽

Number Of Components ◽

Symmetric Extension ◽

Alternative Approach ◽

Tree Traversal ◽

Separable Permutations ◽

Rooted Binary Tree ◽

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving the two quintuples $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{comp},\mathsf{iar})$ have the same distribution over separable permutations. Here for a permutation $\pi$, $\mathrm{LMAX}(\pi)/\mathrm{LMIN}(\pi)$ is the set of values of the left-to-right maxima/minima of $\pi$ and $\mathrm{DESB}(\pi)$ is the set of descent bottoms of $\pi$, while $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are respectively the number of components of $\pi$ and the length of initial ascending run of $\pi$. Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides (up to the classical Knuth–Richards bijection) an alternative approach to a result of Rubey (2016) that asserts the two triples $(\mathrm{LMAX},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathsf{comp},\mathsf{iar})$ are equidistributed on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin–Bagno–Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive. Some equidistribution results for various statistics concerning tree traversal are presented in the end.

On Cyclic Schur-Positive Sets of Permutations

10.37236/8974 ◽

2020 ◽

Vol 27 (2) ◽

Author(s):

Jonathan Bloom ◽

Sergi Elizalde ◽

Yuval Roichman

Keyword(s):

Conjugacy Classes ◽

Young Tableaux ◽

Classical Notion ◽

Descent Set ◽

Standard Young Tableaux ◽

We introduce a notion of cyclic Schur-positivity for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures by the last two authors and by Adin–Gessel–Reiner–Roichman, and yields new examples of Schur-positive sets.

DUAL EQUIVALENCE GRAPHS I: A NEW PARADIGM FOR SCHUR POSITIVITY

Forum of Mathematics Sigma ◽

10.1017/fms.2015.15 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 7

Author(s):

SAMI H. ASSAF

Keyword(s):

Systematic Study ◽

Generating Functions ◽

Universal Method ◽

New Paradigm ◽

Quasisymmetric Functions ◽

Dual Equivalence ◽

Combinatorial Construction ◽

Dual Equivalence Graphs ◽

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Annales Fennici Mathematici ◽

10.5186/aasfm.2021.4642 ◽

2021 ◽

Vol 46 (2) ◽

pp. 633-642

Author(s):

Geraldo Botelho ◽

José Lucas P. Luiz

Keyword(s):

Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

International Mathematics Research Notices ◽

10.1093/imrn/rnz261 ◽

2020 ◽

Author(s):

Sara C Billey ◽

Brendon Rhoades ◽

Vasu Tewari

Keyword(s):

Vector Bundles ◽

Positive Polynomials ◽

Alternating Sign Matrices ◽

Linear Forms ◽

Combinatorial Objects ◽

Theoretic Proof ◽

Boolean Product ◽

Positive Integers ◽

Free Quotient ◽

Abstract Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial $B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.

A generalization of $(q,t)$-Catalan and nabla operators

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3597 ◽

2008 ◽

Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽

Author(s):

N. Bergeron ◽

F. Descouens ◽

M. Zabrocki

Keyword(s):

Schur Functions ◽

Combinatorial Interpretation ◽

Nabla Operator ◽

International Audience ◽

International audience We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths. Nous introduisons des analogues non commutatifs des $k$-fonctions de Schur et nous prouvons que leurs images par l'opérateur nabla non commutatif $\blacktriangledown$ est Schur-rubans positif, à un signe global près. Guidés par ses résultats, nous définissons de nouvelles filtrations des $(q,t)$-nombres de Catalan usuels en calculant l'image de certaines $k$-fonctions de Schur par l'opérateur nabla commutatif $\nabla$. Dans certains cas particuliers, nous donnons une interprétation combinatoire de ces polynômes en termes de chemins de Dyck imbriqués.

Schur-Positivity in a Square

10.37236/3796 ◽

2014 ◽

Vol 21 (3) ◽

Author(s):

Cristina Ballantine ◽

Rosa Orellana

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Difficult Problem ◽

Positivity Criterion ◽

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition $\nu$, we denote by $\nu^c$ its complement in a square partition $(m^m)$. We conjecture a Schur-positivity criterion for symmetric functions of the form $s_{\mu'}s_{\mu^c}-s_{\nu'}s_{\nu^c}$, where $\nu$ is a partition of weight $|\mu|-1$ contained in $\mu$ and the complement of $\mu$ is taken in the same square partition as the complement of $\nu$. We prove the conjecture in many cases.

Schur-positivity via products of grid classes

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6373 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Sergi Elizalde ◽

Yuval Roichman

Keyword(s):

Algebraic Combinatorics ◽

Quasisymmetric Function ◽

Product Operation ◽

International Audience ◽

Sporadic Cases ◽

Broad Framework ◽

Standing Problem ◽

General Method ◽

International audience Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur- positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.

Necessary Conditions for Schur-Maximality

10.37236/7524 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Foster Tom ◽

Stephanie Van Willigenburg

Keyword(s):

Necessary Conditions ◽

Sufficient Conditions ◽

The Difference ◽

A Chain ◽

McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that $B\leq _s A$ if $s_A-s_B$ is Schur-positive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length $a$ or $(a+1)$ for $a\geq 2$. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are minimal. Additionally, we establish two sufficient conditions for the difference of two ribbons to be Schur-positive, which manifest as diagrammatic operations on ribbons. We also deduce two necessary conditions for the difference of two equitable ribbons to be Schur-positive that rely on rows of length $a$ being at the end, or on rows of length $(a+1)$ being evenly distributed.