Irreducible and connected permutations
نویسنده
چکیده
A permutation π of [n] = {1, 2, . . . , n} is irreducible if π([m]) = [m] for no m ∈ [n], m < n, and it is connected if for no interval I ⊂ [n], 2 ≤ |I| ≤ n − 1, the image π(I) is an interval. We review enumeration of irreducible permutations and their appearances in mathematics. Then we enumerate connected permutations. Asymptotically, there are n!/e2 of them and exactly (for n > 2) their number equals 2(−1)n+1 minus the coefficient of xn in the compositional inverse of 1!x+2!x2 + · · ·. We show that their numbers are not P-recursive, are congruent modulo high powers of 2 to 2(−1)n+1, and are congruent modulo 3 to −Cn−1 + (−1)n where Cn is the Catalan number.
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