A117226 Number of permutations of [n] avoiding the consecutive pattern 1243.
1, 1, 2, 6, 23, 110, 630, 4204, 32054, 274914, 2619692, 27459344, 313990182, 3889585408, 51888955808, 741668212080, 11307669002720, 183174676857608, 3141820432768752, 56882461258572976, 1084056190235653304, 21692744773505849952, 454758269790599361968
Offset: 0
Keywords
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, 2011.
- Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see p. 120.
- Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
- Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
- Steven Finch, Pattern-Avoiding Permutations. [Archived copy]
- Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
- Eric Weisstein's World of Mathematics, Pochhammer Symbol.
- Wikipedia, Falling and rising factorials.
Crossrefs
Programs
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Maple
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(b(u-j, o+j-1, 0), j=`if`(t<0, -t, 1)..u)+ add(b(u+j-1, o-j, `if`(t=0, j, -j)), j=1..o)) end: a:= n-> b(n, 0$2): seq(a(n), n=0..25); # Alois P. Heinz, Nov 07 2013 -
Mathematica
A[x_]:=Integrate[AiryAi[ -t],{t,0,x}]; B[x_]:=Integrate[AiryBi[ -t],{t,0,x}]; c=-3^(2/3)*Gamma[2/3]/2; d=-3^(1/6)*Gamma[2/3]/2; a[n_]:=SeriesCoefficient[1/(c*A[x]+d*B[x]+1),{x,0,n}]*n!; Table[a[n],{n,0,10}] (* fixed by Vaclav Kotesovec, Aug 23 2014 *) (* constant d: *) 1/x/.FindRoot[3^(2/3)*Gamma[2/3]/2 * Integrate[AiryAi[-t],{t,0,x}] + 3^(1/6)*Gamma[2/3]/2 * Integrate[AiryBi[-t],{t,0,x}]==1,{x,1},WorkingPrecision->50] (* Vaclav Kotesovec, Aug 23 2014 *)
Formula
a(n) ~ c * d^n * n!, where d = 0.952891423325053197208702817349165942637814..., c = 1.169657787464830219717093446929792145316... . - Vaclav Kotesovec, Aug 23 2014
From Petros Hadjicostas, Nov 01 2019: (Start)
E.g.f.: 1/W(z), where W(z) := 1 + Sum_{n >= 0} (-1)^(n+1)* z^(3*n+1)/(b(n)*(3*n+1)) with b(n) = A176730(n) = (3*n)!/(3^n*(1/3)_n). (Here (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol, or rising factorial, which is denoted by (x)^n in some papers and books.) The function W(z) satisfies the o.d.e. W'''(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W''(0) = 0. [See Theorem 4.3 (Case 1243 with u = 0) in Elizalde and Noy (2003).]
a(n) = Sum_{m = 0..floor((n-1)/3)} (-3)^m * (1/3)_m * binomial(n, 3*m+1) * a(n-3*m-1) for n >= 1 with a(0) = 1. (End)
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