cp's OEIS Frontend

A231211 Number of permutations of [n] avoiding simultaneously consecutive patterns 123, 1432, 2431, and 3421.

%I A231211 #22 Dec 22 2020 06:41:48
%S A231211 1,1,2,5,14,46,177,790,4024,23056,146777,1027850,7852184,64985116,
%T A231211 579191277,5530869310,56336971744,609708912976,6986749484177,
%U A231211 84510154473170,1076016705993704,14385283719409636,201475033030143477,2950048762311387430,45073424916825354064
%N A231211 Number of permutations of [n] avoiding simultaneously consecutive patterns 123, 1432, 2431, and 3421.
%C A231211 Number of permutations of [n] avoiding simultaneously consecutive step patterns up, up and up, down, down.
%H A231211 Alois P. Heinz, <a href="/A231211/b231211.txt">Table of n, a(n) for n = 0..250</a>
%H A231211 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 A231211 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 A231211 a(n) ~ (1+exp(Pi/2)) * (2/Pi)^(n+1) * n!. - _Vaclav Kotesovec_, Aug 28 2014
%e A231211 a(3) = 5: 132, 213, 231, 312, 321.
%e A231211 a(4) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.
%e A231211 a(5) = 46: 13254, 14253, 14352, ..., 54231, 54312, 54321.
%e A231211 a(6) = 177: 132546, 132645, 142536, ..., 654231, 654312, 654321.
%p A231211 b:= proc(u, o, t) option remember; `if`(t=4, 0, `if`(u+o=0, 1,
%p A231211       add(b(u+j-1, o-j, [2, 4, 2][t]), j=1..o)+
%p A231211       add(b(u-j, o+j-1, [1, 3, 4][t]), j=1..u)))
%p A231211     end:
%p A231211 a:= n-> b(n, 0, 1):
%p A231211 seq(a(n), n=0..30);
%p A231211 # second Maple program
%p A231211 n:=40: c[0,0]:=1: for i to n-1 do c[i,0]:=0 end do: for i to n-1 do for j to i do c[i,j] := c[i,j-1] + c[i-1,i-j] + 1 end do end do: 1, seq(c[k, k]/2, k=1..n-1); # _Sergei N. Gladkovskii_, Jul 27 2015
%t A231211 b[u_, o_, t_] := b[u, o, t] = If[t == 4, 0, If[u + o == 0, 1,
%t A231211     Sum[b[u + j - 1, o - j, {2, 4, 2}[[t]]], {j, 1, o}] +
%t A231211     Sum[b[u - j, o + j - 1, {1, 3, 4}[[t]]], {j, 1, u}]]];
%t A231211 a[n_] := b[n, 0, 1];
%t A231211 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)
%Y A231211 Column k=0 of A231210.
%Y A231211 Cf. A049774, A177479.
%K A231211 nonn
%O A231211 0,3
%A A231211 _Alois P. Heinz_, Nov 05 2013