Refined restricted involutions

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