Ratios of Naruse-Newton Coefficients Obtained from Descent Polynomials

TL;DR

This paper investigates ratios of Naruse-Newton coefficients derived from descent polynomials, characterizing their bounds, properties, and explicit formulas for specific shapes, advancing understanding of their combinatorial structure.

Contribution

It provides new characterizations of the ratios of Naruse-Newton coefficients, including bounds, properties, and explicit formulas for ribbons of staircase shape.

Findings

The ratio set R_{a, b} is unbounded above.

Minimal ratios are characterized for certain finite sets.

Explicit formulas are derived for ratios related to staircase-shaped ribbons.

Abstract

We study Naruse-Newton coefficients, which are obtained from expanding descent polynomials in a Newton basis introduced by Jiradilok and McConville. These coefficients $C_0, C_1, \ldots$ form an integer sequence associated to each finite set of positive integers. For fixed nonnegative integers $a<b$, we examine the set $R_{a, b}$ of all ratios $\frac{C_a}{C_b}$ over finite sets of positive integers. We characterize finite sets for which $\frac{C_a}{C_b}$ is minimized and provide a construction to prove $R_{a, b}$ is unbounded above. We use this construction to obtain results on the closure of $R_{a, b}$. We also examine properties of Naruse-Newton coefficients associated with doubleton sets, such as unimodality and log-concavity. Finally, we find an explicit formula for all ratios $\frac{C_a}{C_b}$ of Naruse-Newton coefficients associated with ribbons of staircase shape.