A138162 - OEIS

%I #23 Oct 31 2015 09:15:40

%S 12,96,526,2593,12165,55482,248509,1099255,4817998,20968680,90747564,

%T 390927869,1677551078,7174848666,30598014925,130155932685,

%U 552386655300,2339526458640,9890067346740,41737405295250,175859194700958

%N Number of permutations of {1,2,...,n} containing exactly 4 occurrences of the 132 pattern.

%H Miklós Bóna, <a href="http://dx.doi.org/10.1006/aama.1997.0528">The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!</a>, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.

%H Miklós Bóna, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00062-9">Permutations with one or two 132-subsequences</a>, Discrete Math., 181 (1998) 267-274.

%H T. Mansour and A. Vainshtein, <a href="http://arXiv.org/abs/math.CO/0105073">Counting occurrences of 132 in a permutation</a>, arXiv:math/0105073 [math.CO], 2001.

%F a(n) = (n^9+102n^8-282n^7-12264n^6+32589n^5+891978n^4-7589428n^3 +25452024n^2-39821760n +23950080)(2n-12)!/[24n!(n-6)! ] for n>=6, a(5)=12.

%F G.f.: (1/2)[P(x) + Q(x)/(1-4x)^(7/2)], where P(x)=5x^4-7x^3+2x^2+8x-3, Q(x)=2x^9 +218x^8+1074x^7 -1754x^6 +388x^5 +1087x^4 -945x^3+320x^2-50x+3.

%e a(5)=12 because we have 12534, 12453, 14253, 14523, 13254, 13524, 15324, 14352, 31542, 21534, 21453 and 25143.

%p P:=5*x^4-7*x^3+2*x^2+8*x-3: Q:=2*x^9+218*x^8+1074*x^7-1754*x^6 +388*x^5 +1087*x^4-945*x^3+320*x^2-50*x+3: g:=(P+Q/(1-4*x)^(7/2))*1/2: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=5..25);

%Y Cf. A002054, A082970, A082971, A138163.

%Y Column k=4 of A263771.

%K nonn

%O 5,1

%A _Emeric Deutsch_, Mar 27 2008