Partially 2-Colored Permutations and the Boros-Moll Polynomials

TL;DR

This paper introduces a combinatorial model using partially 2-colored permutations to interpret Boros-Moll polynomial coefficients, providing proofs of recurrence relations and confirming their log-concavity.

Contribution

It offers a novel combinatorial framework for Boros-Moll polynomials, including proofs of recurrence relations and log-concavity.

Findings

Combinatorial interpretation of polynomial coefficients

Proof of recurrence relations for Boros-Moll polynomials

Confirmation of log-concavity conjecture

Abstract

We find a combinatorial setting for the coefficients of the Boros-Moll polynomials $P_m(a)$ in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of $P_m(a)$. This approach enables us to give a combinatorial interpretation of the log-concavity of $P_m(a)$ which was conjectured by Moll and confirmed by Kauers and Paule.