On a particular specialization of monomial symmetric functions

TL;DR

This paper studies a specialized family of monomial symmetric functions deformed by a parameter q, introducing polynomials with integer coefficients, exploring their properties, and conjecturing positivity and log-concavity based on computational evidence and matroid theory.

Contribution

It introduces a new family of polynomials generalizing monomial symmetric functions, establishes their algebraic properties, and conjectures positivity and log-concavity supported by theoretical and computational evidence.

Findings

Polynomials $J_{\lambda}(q)$ have integer coefficients.

Coefficients of $J_{n}^{(r)}$ are positive and log-concave, supported by Huh's theorem.

Last coefficients of $J_{\lambda}$ relate to Pascal's triangle.

Abstract

Let $m_{\lambda }$ be the monomial symmetric functions, $ \lambda $ being an integer partition of $n\in \mathbb{N}^{\ast }$. For the specialization corresponding to the $q$-deformation of the exponential, we prove that each $m_{\lambda }$ is associated with a polynomial $J_{\lambda }\left( q\right) $ whose coefficients belong to $\mathbb{Z}$. $J_{\lambda }$ is a generalization of the case $\lambda =\left( n\right) $ for which $ J_{\left( n\right) }=J_{n}$ is the enumerator of tree inversions. Some relations between $J_{\lambda }$ and $J_{n}^{\left( r\right) }$ are obtained, these $J_{n}^{\left( r\right) }$ having been defined algebraically in a previous work of the author for $n\geq r\geq 1$ and being classically combinatorial enumerators with $J_{n}^{\left( 1\right) }=J_{n}$. From the calculation by induction of $J_{\lambda }$ for $n\leq 6$, we conjecture that the coefficients of each $J_{\lambda }$ are strictly positive and log-concave. As a consequence of Huh's Theorem on the $h$-vector of matroid complex it is shown that the coefficients of $J_{n}^{\left( r\right) }$ are strictly positive and log-concave, which gives a second argument in favor of these conjectures. It is also proven that the last $n-1$ coefficients of $ J_{\lambda }$ are proportional to the first coefficients of column $n-r-1$ of Pascal's triangle, $r$ being the length of $\lambda $. This is a third argument to state the conjectures. The calculation of $J_{\left( 3,2,1\right) }$ shows the existence of an obstacle, if one wants to prove the conjectures by application of Huh's theorem cited above.