A202213 Number of permutations of [n] avoiding the consecutive pattern 45321.

1, 1, 2, 6, 24, 119, 708, 4914, 38976, 347765, 3447712, 37598286, 447294144, 5764747515, 80011430240, 1189835682714, 18873422539776, 318085061976105, 5676223254661760, 106919460527212950, 2119973556022047744, 44136046410218669055, 962630898723772565760

Offset

0,3

Comments

a(n) is the number of permutations on [n] that avoid the consecutive pattern 45321. It is the same as the number of permutations which avoid 12354, 21345 or 54312.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 1..40 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 Theorem 3.2 (p. 116) with m = a = 3 and u = 0.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Wikipedia, Falling and rising factorials.

Formula

From Petros Hadjicostas, Nov 02 2019: (Start)

E.g.f.: 1/W(z), where W(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^(4*n+1)/(b(n)*(4*n+1)) with b(n) = A329070(n,4) = (4*n)!/(4^n*(1/4)_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^(4)(z) + z*W'(z) = 0 with W(0) = 1, W'(0) = -1, and W^(k)(0) = 0 for k = 2..3. [See Theorem 3.2 (with m = a = 3 and u = 0) in Elizalde and Noy (2003).]

a(n) = Sum_{m = 0..floor((n-1)/4)} (-4)^m * (1/4)_m * binomial(n, 4*m+1) * a(n-4*m-1) for n >= 1 with a(0) = 1. (End)

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u+j-1, o-j, `if`(u+j-1<j, 0, j)), j=1..o)+
      `if`(t=-2, 0, add(b(u-j, o+j-1, `if`(j<t, 0,
      `if`(t>0, -1, `if`(t=-1, -2, 0)))), j=1..u)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 19 2013

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[b[u+j-1, o-j, If[u+j-1 < j, 0, j]], {j, 1, o}] + If[t == -2, 0, Sum[b[u-j, o+j-1, If[j<t, 0, If[t>0, -1, If[t == -1, -2, 0]]]], {j, 1, u}]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)

Crossrefs

Cf. A177523, A202214-A202236, A329070.

Column k = 0 of A264781 and row m = 2 of A327722.

Sequence in context: A094198 A297200 A071077 * A202216 A202217 A202221

Adjacent sequences: A202210 A202211 A202212 * A202214 A202215 A202216

Keyword

nonn

Author

Ray Chandler, Dec 14 2011

Status

approved