By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the "Slice and Push Inequality", which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials.
@article{10_37236_10753,
author = {Pakawut Jiradilok and Thomas McConville},
title = {Roots of descent polynomials and an algebraic inequality on hook lengths},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {4},
doi = {10.37236/10753},
zbl = {1532.05175},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10753/}
}
TY - JOUR
AU - Pakawut Jiradilok
AU - Thomas McConville
TI - Roots of descent polynomials and an algebraic inequality on hook lengths
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10753/
DO - 10.37236/10753
ID - 10_37236_10753
ER -
%0 Journal Article
%A Pakawut Jiradilok
%A Thomas McConville
%T Roots of descent polynomials and an algebraic inequality on hook lengths
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10753/
%R 10.37236/10753
%F 10_37236_10753
Pakawut Jiradilok; Thomas McConville. Roots of descent polynomials and an algebraic inequality on hook lengths. The electronic journal of combinatorics, Tome 30 (2023) no. 4. doi: 10.37236/10753
