Ratios of Naruse-Newton Coefficients Obtained from Descent Polynomials

Abstract

We study Naruse-Newton coefficients, which are obtained from expanding descent polynomials in a Newton basis introduced by Jiradilok and McConville. These coefficients C0, C1, . . . 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 Ca C b over finite sets of positive integers. We characterize finite sets for which Ca 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 Ca C b of Naruse-Newton coefficients associated with ribbons of staircase shape.

Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths

“…Remark 4. After the preprint version of this present paper became available, Cai [6] investigated this sequence (C 0 , C 1 , . .…”

“…Cai calls these coefficients "Naruse-Newton coefficients," and examines various properties, such as log-concavity, unimodality, and limiting behavior. In Appendix A of [6], Cai gives tables listing the values of the coefficients for all nonempty descent sets I ⊆ {1, 2, . .…”

“…For readers interested in computational data, we recommend Cai's work [6], especially the table of coefficients in the appendix.…”