A317128 Number of permutations of [n] whose lengths of increasing runs are Fibonacci numbers.

1, 1, 2, 6, 23, 112, 652, 4425, 34358, 299971, 2910304, 31059715, 361603228, 4560742758, 61947243329, 901511878198, 13994262184718, 230811430415207, 4030772161073249, 74301962970014978, 1441745847111969415, 29374226224980834077, 626971133730275593916

Offset: 0

Author

Alois P. Heinz, Jul 21 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000045, A097597, A273001, A317111, A317129, A317130, A317131, A317132, A317444.

Programs

Maple

g:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);

Mathematica

g[n_] := With[{t = 5n^2}, If[IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4], 1, 0]];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)