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A177478 Permutations avoiding the consecutive patterns 4312 and 4213. 4
1, 1, 2, 6, 22, 100, 540, 3388, 24248, 195048, 1742860, 17127880, 183617280, 2132433940, 26669752928, 357375269160, 5108084756320, 77574769941760, 1247401873186560, 21172559509803520, 378282904982091200, 7096584257305845120, 139471475802695196160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) gives the number of permutations of [n] which avoid both the pattern 4312 and 4213 consecutively.  Also the number avoiding the pairs {2134, 3124}, {1243, 1342}, or {3421, 2431} (by symmetry).

This can also be considered avoiding a partially ordered pattern:  Suppose p<s, r<s, o<p and o<r. To avoid spor means not to have four consecutive letters such that the first letter is larger than the second and the last one, the third letter is less than the second and the last one.

The Baxter-Nakamura-Zeilberger paper has an associated Maple package.  See Links.

LINKS

Ray Chandler and Alois P. Heinz, Table of n, a(n) for n = 0..120 (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

S. Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

FORMULA

a(n) ~ c * d^n * n!, where d = 0.89333294588184091624317413051..., c = 1.4839698712287023868073431417... . - Vaclav Kotesovec, Aug 24 2014

MAPLE

b:= proc(u, o, s, t) option remember; `if`(u+o=0, 1,

       add(b(u-j, o+j-1, t, j), j=1..u)+

       add(b(u+j-1, o-j, 0, 0), j=`if`(s>0, s+t-1, 1)..o))

    end:

a:= n-> b(0, n, 0, 0):

seq(a(n), n=0..25);  # Alois P. Heinz, Oct 25 2013

MATHEMATICA

b[u_, o_, s_, t_] := b[u, o, s, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, t, j], {j, 1, u}] + Sum[b[u+j-1, o-j, 0, 0], {j, If[s > 0, s+t-1, 1], o}]];

a[n_] := b[0, n, 0, 0];

a /@ Range[0, 25] (* Jean-Fran├žois Alcover, Nov 03 2020, after Alois P. Heinz *)

CROSSREFS

Cf. A117226, A117156.

Sequence in context: A012266 A009468 A088819 * A052517 A245119 A012270

Adjacent sequences:  A177475 A177476 A177477 * A177479 A177480 A177481

KEYWORD

nonn

AUTHOR

Signy Olafsdottir (signy06(AT)ru.is), May 09 2010

EXTENSIONS

More terms, succinct title, additional comments, new references from Andrew Baxter, Jan 21 2011

STATUS

approved

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Last modified April 22 07:26 EDT 2021. Contains 343163 sequences. (Running on oeis4.)