Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Refined Restricted Involutions   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 $i
\geq 2$, and define $I_n^k(\emptyset;\alpha)$ 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 $i \geq 2$, exactly once. Let $i_n^k(\alpha)$ be the number
of elements in $I_n^k(\alpha)$ and let $i_n^k(\emptyset;\alpha)$ be the number
of elements in $I_n^k(\emptyset;\alpha)$. We investigate $I_n^k(\alpha)$ and
$I_n^k(\emptyset;\alpha)$ for all $\alpha \in S_3$. In particular, we show that
$i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$,
$i_n^k(\emptyset;132) =i_n^k(\emptyset;213)$, and
$i_n^k(\emptyset;231)=i_n^k(\emptyset;312)$ for all $0 \leq k \leq n$.

24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn