A071076 Number of permutations that avoid the generalized pattern 123-4.

1, 1, 2, 6, 23, 108, 598, 3815, 27532, 221708, 1970251, 19150132, 202064380, 2300071071, 28092017668, 366425723926, 5083645400819, 74745472084176, 1160974832572274, 18995175706664735, 326531476287842760, 5883736110875887560, 110893188848753125475

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.

Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011.

Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Formula

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = (sqrt(3)/2)*exp(y/2)/cos((sqrt(3)/2)*y + Pi/6).

Let b(n) = A049774(n) = number of permutations of [n] that avoid the consecutive pattern 123. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] -

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
      `if`(t=1 and o>j, 0, b(u+j-1, o-j, t+1)), j=1..o)+
       add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 14 2015

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == 1 && o > j, 0, b[u + j - 1, o - j, t + 1]], {j, 1, o}] + Sum[b[u - j, o + j - 1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 01 2016, after Alois P. Heinz *)

Crossrefs

Cf. A049774, A071088, A071075, A071077.

Sequence in context: A101053 A155857 A378734 * A297196 A112501 A093345

Adjacent sequences: A071073 A071074 A071075 * A071077 A071078 A071079

Keyword

nonn

Author

Sergey Kitaev, May 26 2002

Extensions

More terms from Vladeta Jovovic, May 28 2002

Link added by Andrew Baxter, May 17 2011

Typos in formula corrected by Vaclav Kotesovec, Aug 23 2014

Status

approved