A317129 Number of permutations of [n] whose lengths of increasing runs are squares.

1, 1, 1, 1, 2, 9, 40, 151, 571, 2897, 19730, 140190, 953064, 6708323, 54631552, 510143776, 4987278692, 49168919669, 505209884549, 5638095015594, 67921924172174, 852861260421398, 10992380368532792, 147296144926635359, 2082906807168675698, 30973237281668975230

Offset: 0

Author

Alois P. Heinz, Jul 21 2018

Keywords

• nonn

Examples

			a(3) = 1: 321.
a(4) = 2: 1234, 4321.
a(5) = 9: 12354, 12453, 13452, 21345, 23451, 31245, 41235, 51234, 54321.
		

Links

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

Crossrefs

Cf. A000290, A097597, A205801, A317111, A317128, A317130, A317131, A317132, A317445.

Programs

Maple

g:= n-> `if`(issqr(n), 1, 0):
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_] := If[IntegerQ@Sqrt[n], 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 *)