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arXiv · 2301.00341

Self-reciprocal polynomials connecting unsigned and signed relative derangements

Ricky X. F. Chen·Yu-Chen Ruan
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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.

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Ricky X. F. Chen, Yu-Chen Ruan. 2023-01-01. Self-reciprocal polynomials connecting unsigned and signed relative derangements. https://doi.org/10.1080/10236198.2023.2253331

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