Self-reciprocal polynomials connecting unsigned and signed relative derangements

TL;DR

This paper introduces symmetric polynomials for signed relative derangements, revealing their recursive structure, unimodality, and connections between signed and unsigned derangements, with combinatorial proofs and new relations.

Contribution

It defines and analyzes polynomials linking signed and unsigned relative derangements, providing recursive formulas, unimodality results, and a dual relation, advancing combinatorial understanding.

Findings

Polynomials are symmetric and connect signed with unsigned derangements.

Coefficients satisfy a specific recursion and are unimodal.

New dual relation between signed and relative derangements is established.

Abstract

In this paper, we introduce polynomials (in $t$) of signed relative derangements that track the number of signed elements. The polynomials are clearly seen to be in a sense symmetric. Note that relative derangements are those without any signed elements, i.e., the evaluations of the polynomials at $t=0$. Also, the numbers of all signed relative derangements are given by the evaluations at $t=1$. Then the coefficients of the polynomials connect unsigned and signed relative derangements and show how putting elements with signs affects the formation of derangements. We first prove a recursion satisfied by these polynomials which results in a recursion satisfied by the coefficients. A combinatorial proof of the latter is provided next. We also show that the sequences of the coefficients are unimodal. Moreover, other results are obtained. For instance, a kind of dual of a relation between signed derangements and signed relative derangements previously proved by Chen and Zhang is presented.