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A056986

Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.

203

0, 0, 1, 10, 78, 588, 4611, 38890, 358018, 3612004, 39858014, 478793588, 6226277900, 87175616760, 1307664673155, 20922754530330, 355687298451210, 6402373228089300, 121645098641568810, 2432902001612519580, 51090942147243172980, 1124000727686125116360

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OFFSET

1,4

COMMENTS

This is well-defined because for all patterns alpha in S_3 the number of permutations in S_n avoiding alpha is the same (the Catalan numbers). - Emeric Deutsch, May 05 2008

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..170

FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3, The number of occurrences of the pattern [1,3,2] in a permutation, The number of occurrences of the pattern [2,1,3] in a permutation, The number of occurrences of the pattern [2,3,1] in a permutation, The number of occurrences of the pattern [3,1,2] in a permutation

R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, pp. 383-406, 1985.

Eric Weisstein's World of Mathematics, Permutation Pattern

FORMULA

From Alois P. Heinz, Jul 05 2012: (Start)

a(n) = A214152(n, 3).

a(n) = A000142(n) - A000108(n).

a(n) = A000142(n) - A214015(n, 2). (End)

E.g.f.: 1/(1 - x) - exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - Ilya Gutkovskiy, Jan 21 2017

EXAMPLE