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A231166 Number of permutations of [n] avoiding simultaneously consecutive patterns 1243, 1342, and 1324.

%I A231166 #18 Feb 27 2020 07:40:49
%S A231166 1,1,2,6,21,91,467,2755,18523,139740,1169616,10763807,108028386,
%T A231166 1174391384,13748315494,172439034531,2306986699190,32792999417180,
%U A231166 493559520202535,7841127918788283,131127477517244419,2302491655047553206,42355105188617740229
%N A231166 Number of permutations of [n] avoiding simultaneously consecutive patterns 1243, 1342, and 1324.
%H A231166 Alois P. Heinz, <a href="/A231166/b231166.txt">Table of n, a(n) for n = 0..70</a>
%H A231166 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>, 2011.
%H A231166 S. Kitaev and T. Mansour, <a href="https://arxiv.org/abs/math/0209340v1">On multi-avoidance of generalized patterns</a>, arXiv:math/0209340 [math.CO], 2002.
%e A231166 a(4) = 24 - 3 = 21 because 1243, 1342, 1324 are avoided.
%p A231166 b:= proc(u, o, s, t) option remember; `if`(u+o=0, 1,
%p A231166        add(b(u-j, o+j-1, `if`(t>0, t, 0), `if`(t>0, -j, 0)),
%p A231166            j=`if`(s>0 and t>0,s+t-1,1)..u)+
%p A231166        add(b(u+j-1, o-j, `if`(t>0, t, 0), +j),
%p A231166            j=1..`if`(s>0 and t<0 and -t<s, -t-1, o)))
%p A231166     end:
%p A231166 a:= n-> b(n, 0$3):
%p A231166 seq(a(n), n=0..25);
%t A231166 b[u_, o_, s_, t_] := b[u, o, s, t] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, If[t > 0, t, 0], If[t > 0, -j, 0]], {j, If[s > 0 && t > 0, s + t - 1, 1], u}] + Sum[b[u + j - 1, o - j, If[t > 0, t, 0], +j], {j, 1, If[s > 0 && t < 0 && -t < s, -t - 1, o]}]];
%t A231166 a[n_] := b[n, 0, 0, 0];
%t A231166 a /@ Range[0, 25] (* _Jean-François Alcover_, Feb 27 2020, after _Alois P. Heinz_ *)
%Y A231166 Cf. A113228, A117156, A117226.
%K A231166 nonn
%O A231166 0,3
%A A231166 _Alois P. Heinz_, Nov 04 2013