Enumerating pairs of permutations with the same up-down form |
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Institution: | Bell Laboratories, Murray Hill, NJ 07974, USA |
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Abstract: | For each sequence q = {qi} = ± 1, i = 1, …, n−1 let Nq = the number of permutations σ of 1, 2, …, n with up-down sequence sgn(σi+1 − σi) = qi, i = 1,…, n−1. Clearly Σq (Nq/n!) =1 but what is the probability pn = Σq (Nq/n!)2 that two random permutations have the same up-down sequence? We show that pn = (Kn−11,1) where 1 = 1(x, y) ≡ 1 and Kn−1 is the iterated integral operator with Kφ(x, y) = ∫01∫01 K(x, y; x′, y′)φ(x′, y′) dx′ dy′ on L20, 1] × 0, 1] where K(x, y; x′, y′) is 1 if (x− x′)(y − y′) > 0 otherwise, and (f, g) = ∫01∫01fg. The eigenexpression of K yeilds pn ∼ cαn as n → ∞, where c ≈ 1.6, α ≈ 0.55.We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms. |
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