Refined restricted involutions

Define Ink(α) to be the set of involutions of {1,2,…,n}{1,2,…,n} with exactly kk fixed points which avoid the pattern α∈Siα∈Si, for some i≥2i≥2, and define Ink(0̸;α) to be the set of involutions of {1,2,…,n}{1,2,…,n} with exactly kk fixed points which contain the pattern α∈Siα∈Si, for some i≥2i≥2, exactly once. Let ink(α) be the number of elements in Ink(α) and let ink(0̸;α) be the number of elements in Ink(0̸;α). We investigate Ink(α) and Ink(0̸;α) for all α∈S3α∈S3. In particular, we show that ink(132)=ink(213)=ink(321), ink(231)=ink(312), ink(0̸;132)=ink(0̸;213), and ink(0̸;231)=ink(0̸;312) for all 0≤k≤n0≤k≤n.