A056986 Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.

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

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

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

Examples

			a(4) = 10 because, taking, for example, the pattern alpha=321, we have 3214, 3241, 1432, 2431, 3421, 4213, 4132, 4231, 4312 and 4321.
		

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

Crossrefs

Cf. A000108, A000142, A138159, A214015, A214152.

Programs

Magma

A056986:= func< n | Factorial(n) - Catalan(n) >;
[A056986(n): n in [1..30]]; // G. C. Greubel, Oct 06 2024

Maple

a:= n-> n! -binomial(2*n, n)/(n+1):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 05 2012

Mathematica

Table[n! -CatalanNumber[n], {n,30}]

PARI

a(n)=n!-binomial(n+n,n+1)/n \\ Charles R Greathouse IV, Jun 10 2011

SageMath

def A056986(n): return factorial(n) - catalan_number(n)
[A056986(n) for n in range(1,31)] # G. C. Greubel, Oct 06 2024

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