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%I #28 Jan 21 2017 15:58:11
%S 0,0,1,10,78,588,4611,38890,358018,3612004,39858014,478793588,
%T 6226277900,87175616760,1307664673155,20922754530330,355687298451210,
%U 6402373228089300,121645098641568810,2432902001612519580,51090942147243172980,1124000727686125116360
%N Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.
%C 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
%H Alois P. Heinz, <a href="/A056986/b056986.txt">Table of n, a(n) for n = 1..170</a>
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000002/">The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000220/">The number of occurrences of the pattern [1,3,2] in a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000218/">The number of occurrences of the pattern [2,1,3] in a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000219/">The number of occurrences of the pattern [2,3,1] in a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000217/">The number of occurrences of the pattern [3,1,2] in a permutation</a>
%H R. Simion and F. W. Schmidt, <a href="http://dx.doi.org/10.1016/S0195-6698(85)80052-4">Restricted permutations</a>, European J. Combin., 6, pp. 383-406, 1985.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationPattern.html">Permutation Pattern</a>
%F a(n) = A214152(n,3) = A000142(n)-A000108(n) = A000142(n)-A214015(n,2). - _Alois P. Heinz_, Jul 05 2012
%F E.g.f.: 1/(1 - x) - exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - _Ilya Gutkovskiy_, Jan 21 2017
%e a(4) = 10 because, taking, for example, the pattern alpha=321, we have 3214, 3241, 1432, 2431, 3421, 4213, 4132, 4231, 4312 and 4321.
%p a:= n-> n! -binomial(2*n, n)/(n+1):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Jul 05 2012
%t n!-Binomial[2n, n]/(n+1)
%o (PARI) a(n)=n!-binomial(n+n,n+1)/n \\ _Charles R Greathouse IV_, Jun 10 2011
%Y Cf. A000108, A000142, A138159, A214015, A214152.
%K nonn,easy
%O 1,4
%A _Eric W. Weisstein_
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