A138162 Number of permutations of {1,2,...,n} containing exactly 4 occurrences of the 132 pattern.

12, 96, 526, 2593, 12165, 55482, 248509, 1099255, 4817998, 20968680, 90747564, 390927869, 1677551078, 7174848666, 30598014925, 130155932685, 552386655300, 2339526458640, 9890067346740, 41737405295250, 175859194700958

Offset

5,1

Links

Table of n, a(n) for n=5..25.

Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.

Miklós Bóna, Permutations with one or two 132-subsequences, Discrete Math., 181 (1998) 267-274.

T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Formula

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.

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.

Example

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

Maple

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);

Crossrefs

Cf. A002054, A082970, A082971, A138163.

Column k=4 of A263771.

Sequence in context: A341233 A120658 A121627 * A264418 A073392 A038845

Adjacent sequences: A138159 A138160 A138161 * A138163 A138164 A138165

Keyword

nonn

Author

Emeric Deutsch, Mar 27 2008

Status

approved