A117156 Number of permutations avoiding the consecutive pattern 1342.

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)

A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

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 *)

Crossrefs

Cf. A113228, A113229, A117158, A117226, A201692, A201693, A231166.

Cf. A052324.

Sequence in context: A320566 A205802 A117226 * A201692 A113229 A113228

Adjacent sequences: A117153 A117154 A117155 * A117157 A117158 A117159

Keyword

nonn

Author

Steven Finch, Apr 26 2006

Status

approved