A Duality between Hilbert Functions of Lex Ideals and Quotients

TL;DR

This paper explores a duality between Hilbert functions of lex ideals and their quotients, providing explicit formulas for Macaulay coefficients and revealing a set partition structure.

Contribution

It introduces a novel duality linking ideal and quotient Macaulay coefficients with explicit formulas and a set partition characterization.

Findings

Established a duality between ideal and quotient Macaulay coefficients.

Derived explicit formulas for these coefficients.

Proved that the coefficients form a set partition of a specific integer set.

Abstract

We study the Macaulay coefficients induced by the ideal and quotient segments of a degree-$\delta$ monomial in $n$ variables. We give explicit formulas for these coefficients and establish a duality between the two theories. Our main result is that the ideal and quotient coefficients form a set partition of $\{0,1,\ldots,n+\delta-2\}$.