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

1, 1, 2, 6, 21, 91, 467, 2755, 18523, 139740, 1169616, 10763807, 108028386, 1174391384, 13748315494, 172439034531, 2306986699190, 32792999417180, 493559520202535, 7841127918788283, 131127477517244419, 2302491655047553206, 42355105188617740229

Offset: 0

Author

Alois P. Heinz, Nov 04 2013

Keywords

• nonn

Examples

			a(4) = 24 - 3 = 21 because 1243, 1342, 1324 are avoided.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..70
• A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
• S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, arXiv:math/0209340 [math.CO], 2002.

Crossrefs

Cf. A113228, A117156, A117226.

Programs

Maple

b:= proc(u, o, s, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, `if`(t>0, t, 0), `if`(t>0, -j, 0)),
           j=`if`(s>0 and t>0,s+t-1,1)..u)+
       add(b(u+j-1, o-j, `if`(t>0, t, 0), +j),
           j=1..`if`(s>0 and t<0 and -t b(n, 0$3):
seq(a(n), n=0..25);

Mathematica

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]}]];
a[n_] := b[n, 0, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz *)