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A231385 Number of permutations of [n] avoiding simultaneously consecutive step patterns UUD, UDU, DUU (U=up, D=down).

%I A231385 #14 Dec 22 2020 06:41:44
%S A231385 1,1,2,6,13,39,158,674,3304,19511,122706,834131,6416525,52909708,
%T A231385 462097526,4395014406,44626369587,476351029850,5414386451909,
%U A231385 65177788719791,821378978885730,10880928171304446,151423268838929524,2197946731864495343,33278572455563069142
%N A231385 Number of permutations of [n] avoiding simultaneously consecutive step patterns UUD, UDU, DUU (U=up, D=down).
%C A231385 The avoided patterns are: 1243, 1342, 2341 (=UUD), 1324, 1423, 2314, 2413, 3412 (=UDU), 2134, 3124, 4123 (=DUU).
%H A231385 Alois P. Heinz, <a href="/A231385/b231385.txt">Table of n, a(n) for n = 0..200</a>
%H A231385 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 A231385 S. Kitaev and T. Mansour, <a href="http://www.ru.is/kennarar/sergey/index_files/Papers/multi_avoid_gen_patterns.pdf">On multi-avoidance of generalized patterns</a>
%F A231385 a(n) ~ c * d^n * n!, where d = 0.63140578989563018836..., c = 3.3290259175437715006... . - _Vaclav Kotesovec_, Aug 28 2014
%e A231385 a(4) = 13: 1234, 1432, 2143, 2431, 3142, 3214, 3241, 3421, 4132, 4213, 4231, 4312, 4321.
%e A231385 a(5) = 39: 12345, 14325, 15324, ..., 54231, 54312, 54321.
%e A231385 a(6) = 158: 123456, 143265, 153264, ..., 654231, 654312, 654321.
%p A231385 b:= proc(u, o, t) option remember; `if`(t=7, 0, `if`(u+o=0, 1,
%p A231385       add(b(u+j-1, o-j, [2, 3, 3, 6, 7, 7][t]), j=1..o)+
%p A231385       add(b(u-j, o+j-1, [4, 5, 7, 4, 4, 5][t]), j=1..u)))
%p A231385     end:
%p A231385 a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 1), j=1..n)):
%p A231385 seq(a(n), n=0..25);
%t A231385 b[u_, o_, t_] := b[u, o, t] = If[t == 7, 0, If[u + o == 0, 1,
%t A231385     Sum[b[u + j - 1, o - j, {2, 3, 3, 6, 7, 7}[[t]]], {j, 1, o}] +
%t A231385     Sum[b[u - j, o + j - 1, {4, 5, 7, 4, 4, 5}[[t]]], {j, 1, u}]]];
%t A231385 a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, 1], {j, 1, n}]];
%t A231385 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)
%Y A231385 Column k=0 of A231384.
%K A231385 nonn
%O A231385 0,3
%A A231385 _Alois P. Heinz_, Nov 08 2013