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%I #27 Apr 08 2021 12:43:33
%S 1,1,2,4,10,30,108,454,2186,11840,71254,471620,3405444,26638846,
%T 224412432,2025537526,19501264282,199486609542,2160670133172,
%U 24702689130878,297287351106034,3756625585698012,49730495350888890,688260442829860292,9939538985274953060
%N The number of permutations avoiding simultaneously consecutive patterns 213 and 231.
%D S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, Ars Combinatoria 76 (2005), 321-350.
%H Alois P. Heinz, <a href="/A173940/b173940.txt">Table of n, a(n) for n = 0..200</a>
%H R. E. L. Aldred, M. D. Atkinson, and D. J. McCaughan, <a href="https://doi.org/10.1016/j.aam.2010.03.005">Avoiding consecutive patterns in permutations</a> Adv. in Appl. Math., 45(3), 449-461, 2010.
%H A. Baxter, B. Nakamura, and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>
%H S. Kitaev, <a href="https://web.archive.org/web/20061013131319/http://www.ru.is/kennarar/sergey/index_files/Papers/multi_avoid_gen_patterns.pdf">On multi-avoidance of generalized patterns</a>
%F 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.
%F a(n) ~ c * d^n * n!, where d = 0.6017307279439431870308246204583329377..., c = 3.15506287220052225202744035520726815... . - _Vaclav Kotesovec_, Aug 22 2014
%e For n = 3 c(3) = 4 since 123, 132, 312, and 321 are the 3-permutations avoiding 213 and 231.
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, -j), j=1..`if`(t>0, t-1, u))+
%p add(b(u+j-1, o-j, j), j=1..`if`(t<0, -t-1, o)))
%p end:
%p a:= n-> b(n, 0, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 04 2013
%t 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]}]];
%t a[n_] := b[n, 0, 0];
%t a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 21 2020, after _Alois P. Heinz_ *)
%K nonn
%O 0,3
%A Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
%E More terms from _Alois P. Heinz_, Nov 03 2013
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