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A117156
Number of permutations avoiding the consecutive pattern 1342.
18
1, 1, 2, 6, 23, 110, 630, 4210, 32150, 276210, 2636720, 27687440, 317169270, 3936056080, 52603684760, 753241509900, 11504852242400, 186705357825800, 3208160592252000, 58188413286031600, 1110946958902609400
OFFSET
0,3
COMMENTS
a(n) is the number of permutations on [n] that avoid the consecutive pattern 1342. It is the same as the number of permutations which avoid 2431, 4213, 3124, 1432, 2341, 4123 or 3214.
REFERENCES
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006) 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003) 110-125.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..60 from Ray Chandler)
Steven Finch, Pattern-Avoiding Permutations [Broken link?]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
FORMULA
a(n) ~ c * d^n * n!, where d = 1/r = 0.9546118344740519430556804334164431663486451742931588346372174751881329..., where r = 1.04754620033697244977759528695194261... is the root of the equation integral_{x,0,r} exp(-x^3/6) dx = 1, and c = 1.1561985648406071020520797542907648300587978482957199521032311960968187467... . - Vaclav Kotesovec, Aug 23 2014
MATHEMATICA
a[n_]:=Coefficient[Series[1/(1-Integrate[Exp[ -t^3/6], {t, 0, x}]), {x, 0, 30}], x^n]*n!
(* Second program: *)
m = 21; gf = 1/(1-Sum[If[Mod[k, 3] == 0, (-1)^Mod[k, 6]/(6^(k/3)*(k/3)!), 0]* (x^(k+1)/(k+1)), {k, 0, m}]);
CoefficientList[gf + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, May 11 2019 *)
KEYWORD
nonn
AUTHOR
Steven Finch, Apr 26 2006
STATUS
approved