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A173940 The number of permutations avoiding simultaneously consecutive patterns 213 and 231. 1
1, 1, 2, 4, 10, 30, 108, 454, 2186, 11840, 71254, 471620, 3405444, 26638846, 224412432, 2025537526, 19501264282, 199486609542, 2160670133172, 24702689130878, 297287351106034, 3756625585698012, 49730495350888890, 688260442829860292, 9939538985274953060 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, Ars Combinatoria 76 (2005), 321-350.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

R. E. L. Aldred, M. D. Atkinson, and D. J. McCaughan, Avoiding consecutive patterns in permutations Adv. in Appl. Math., 45(3), 449-461, 2010.

A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

S. Kitaev, On multi-avoidance of generalized patterns

FORMULA

We have c(n)= )=\sum_{i,j=1}^{n} c(n;i,j) with c(n;i;i)=0 for all n, i >= 1; b(n;i,j)= \sum_{k=1}^{i-1}b(n-1;j,k) if i > j; b(n;i,j)= \sum_{k=1}^{i-1}b(n-1;j-1,k) +\sum_{k=j}^{n-1}b(n-1;j-1,k) if i > j; and b(2;1,2)= b(2;2,1)=1, b(2;1,1)= b(2;1,1)=0.

a(n) ~ c * d^n * n!, where d = 0.6017307279439431870308246204583329377..., c = 3.15506287220052225202744035520726815... . - Vaclav Kotesovec, Aug 22 2014

EXAMPLE

For n = 3 c(3) = 4 since 123, 132, 312, and 321 are the 3-permutations avoiding 213 and 231.

MAPLE

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

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

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

    end:

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

seq(a(n), n=0..25);  # Alois P. Heinz, Nov 04 2013

MATHEMATICA

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

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

a /@ Range[0, 25] (* Jean-Fran├žois Alcover, Nov 21 2020, after Alois P. Heinz *)

CROSSREFS

Sequence in context: A091174 A005193 A317280 * A101901 A124384 A280432

Adjacent sequences:  A173937 A173938 A173939 * A173941 A173942 A173943

KEYWORD

nonn

AUTHOR

Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010

EXTENSIONS

More terms from Alois P. Heinz, Nov 03 2013

STATUS

approved

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Last modified June 19 05:18 EDT 2021. Contains 345125 sequences. (Running on oeis4.)