A317445 Number of permutations of [n] whose lengths of increasing runs are distinct squares.

1, 1, 0, 0, 1, 8, 0, 0, 0, 1, 18, 0, 0, 1428, 47998, 0, 1, 32, 0, 0, 9688, 505056, 0, 0, 0, 4085949, 284958912, 0, 0, 290824632172, 28643427712626, 0, 0, 0, 104902510, 9998016202, 1, 72, 23207824626842, 3008268832634364, 182778, 206173972520, 24290829974718, 0

Offset: 0

Author

Alois P. Heinz, Jul 28 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000290, A001422, A003995, A317129, A317444, A317446, A317447, A317448.

Programs

Maple

g:= (n, s)-> `if`(n in s or not issqr(n), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..50);

Mathematica

g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[n], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 24 2021, after Alois P. Heinz *)

Formula

a(n) = 0 <=> n in { A001422 }.

a(n) > 0 <=> n in { A003995 }.