Refined Restricted Involutions

Abstract

Define $I_n^k(\alpha )$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_i$, for some $i\ge 2$, and define $I_n^k(\varnothing ;\alpha )$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $\alpha \in S_i$, for some $i\ge 2$, exactly once. Let $i_n^k(\alpha )$ be the number of elements in $I_n^k(\alpha )$ and let $i_n^k(\varnothing ;\alpha )$ be the number of elements in $I_n^k(\varnothing ;\alpha )$. We investigate $I_n^k(\alpha )$ and $I_n^k(\varnothing ;\alpha )$ for all $\alpha \in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$, $i_n^k(\varnothing ;132)=i_n^k(\varnothing ;213)$, and $i_n^k(\varnothing ;231)=i_n^k(\varnothing ;312)$ for all $0\le k\le n$.

20 pages

References in corpus (1)

• Some statistics on restricted 132 involutions