Symmetric polynomials connecting unsigned and signed relative derangements

Ricky Xiao-Feng Chen, Yu-Chen Ruan

Published 2023-01-01Version 1

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.