Several combinatorial inequalities related to squarefree monomial ideals

TL;DR

This paper establishes new combinatorial inequalities related to squarefree monomial ideals by leveraging the relationship between Hilbert depth and Stanley depth, involving coefficients of a specific polynomial.

Contribution

It introduces several novel combinatorial inequalities involving coefficients of (1+t+...+t^{m-1})^n for squarefree monomial ideals.

Findings

Derived inequalities involving polynomial coefficients

Bounded Stanley depth using Hilbert depth

Enhanced understanding of monomial ideal combinatorics

Abstract

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$, the ring of polynomials in $n$ variables, over $K$. Using the fact that the Hilbert depth is an upper bound for the Stanley depth of a quotient of squarefree monomial ideals $0\subset I\subsetneq J\subset S$, we prove several combinatorial inequalities which involve the coefficients of the polynomial $f(t)=(1+t+\cdots+t^{m-1})^n$.