Counting covered fixed points and covered arcs in an involution

Abstract

An involution on [n] = {1, 2,…, n} is a permutation π ₁ π ₂… π n on [n] such that π _πi = i for all i ∈ [n]. Let π = π ₁ π ₂ n be any involution on [n]. We say that i is a covered fixed point of π if π _i = i and there exists a, b such that 1 ≤ a < i < b ≤ n and π _a = b. We say that ab is a covered arc of π if there exists c, d such that 1 ≤ c < a < b < d ≤ n, π _a = b, and π _c = d. In this paper, we study the generating function for the number of involutions on [n] according to the number of covered fixed points/covered arcs.