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A082971
Number of permutations of {1,2,...,n} containing exactly 3 occurrences of the 132 pattern.
5
1, 14, 82, 410, 1918, 8657, 38225, 166322, 716170, 3059864, 12994936, 54924212, 231235054, 970347575, 4060697955, 16952812170, 70629116910, 293720506860, 1219498444500, 5055891511980, 20933654593020, 86571545598642, 357628915621698, 1475896409177780
OFFSET
4,2
LINKS
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) = (2*n-9)!/n!/6/(n-5)! *(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n +20160).
a(n) = (n^6 + 51n^5 - 407n^4 - 99n^3 + 7750n^2 - 22416n + 20160)(2n-9)!/[6 n!(n-5)! for n>=5; a(4)=1. G.f.=(1/2)(P(x) + Q(x)/(1-4x)^(5/2), where P(x)=2x^3 - 5x^2 + 7x - 2, Q(x)=-22x^6 - 106x^5 + 292x^4 - 302x^3 + 135x^2 - 27x + 2. - Emeric Deutsch, Mar 27 2008
EXAMPLE
a(4)=1 because we have 1432 (the 132 occurrences are 143, 142 and 132).
MAPLE
P:=2*x^3-5*x^2+7*x-2: Q:=-22*x^6-106*x^5+292*x^4-302*x^3+135*x^2-27*x+2: g:= (P+Q/(1-4*x)^(5/2))*1/2: gser:=series(g, x=0, 30): seq(coeff(gser, x, n), n=4..25); # Emeric Deutsch, Mar 27 2008
MATHEMATICA
a[4] = 1; a[n_] := (n^6 + 51 n^5 - 407 n^4 - 99 n^3 + 7750 n^2 - 22416 n + 20160) (2 n - 9)!/(6 n! (n - 5)!);
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Oct 30 2018 *)
PROG
(PARI) a(n)=(2*n-9)!/n!/6/(n-5)!*(n^6+51*n^5-407*n^4-99*n^3 +7750*n^2 -22416*n+20160)
(Magma) [1] cat [(n^6+51*n^5-407*n^4-99*n^3+7750*n^2-22416*n+20160)* Factorial(2*n-9)/(6*Factorial(n)*Factorial(n-5)): n in [5..30]]; // Vincenzo Librandi, Oct 30 2018
CROSSREFS
Column k=3 of A263771.
Sequence in context: A209942 A215700 A199912 * A374650 A176010 A250562
KEYWORD
nonn
AUTHOR
Benoit Cloitre, May 27 2003
EXTENSIONS
Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar
STATUS
approved