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

Author

Steven Finch, Apr 26 2006

Keywords

• nonn

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]

Crossrefs

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

Cf. A052324.

Programs

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 *)

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