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Restricted 132-Involutions and Chebyshev Polynomials

O. Guibert·T. Mansour
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Abstract

We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and $12... k$ have the same enumerative formula according to the length than involutions avoiding both 132 and any {\em double-wedge pattern} possibly followed by fixed points of total length $k$. Many results are also shown with a combinatorial point of view.

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O. Guibert, T. Mansour. 2002-01-15. Restricted 132-Involutions and Chebyshev Polynomials. https://arxiv.org/abs/math/0201136

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