The number of orthogonal permutations

A problem on maximal clones in universal algebra leads to the natural concept of orthogonal orders and their characterization. Two (partial) orders on the same set P are orthogonal if they share only trivial endomorphisms, i.e. if the identity self-map of P is the sole non-constant self-map preserving (i.e. compatible with) both orders. We start with a neat and easy characterization of orthogonal pairs of chains (i.e. linear or total orders) and then proceed to the study of the number q(k) of chains on {0, 1, …, k − 1} orthogonal to the natural chain 0 < 1 < ⋯ < k − 1. We obtain a recurrence formula for q(k) and prove that the ratio q(k)k! (of such chains among all chains) goes to e−2 = 0.1353 ⋯ as k → ∞. Results are formulated in terms of permutations.