Refined Restricted Involutions

الأصل · EN

Define Iₙᵏ(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 Iₙᵏ(;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 iₙᵏ(α) be the number of elements in Iₙᵏ(α) and let iₙᵏ(;α) be the number of elements in Iₙᵏ(;α). We investigate Iₙᵏ(α) and Iₙᵏ(;α) for all α∈ S₃. In particular, we show that iₙᵏ(132)=iₙᵏ(213)=iₙᵏ(321), iₙᵏ(231)=iₙᵏ(312), iₙᵏ(;132) =iₙᵏ(;213), and iₙᵏ(;231)=iₙᵏ(;312) for all 0 ≤ k ≤ n.