A317165 Number of permutations of [n*(n+1)/2] with distinct lengths of increasing runs.

1, 1, 5, 241, 188743, 2734858573, 892173483721887, 7469920269852025033699, 1841449549508718383891930251607, 14973026148724796464136435753195418043885, 4467880642339303169146446437381463615730321314015457, 53810913396105573079543194840166969124601447333276658546225661505

Offset: 0

Author

Alois P. Heinz, Jul 23 2018

Keywords

• nonn

Crossrefs

Cf. A000217, A246292, A317130, A317166, A317273.

Programs

Maple

g:= (n, s)-> `if`(n in s, 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*(n+1)/2, 0$2, {}):
seq(a(n), n=0..8);

Mathematica

g[n_, s_] := If[MemberQ[s, 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, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];
a[n_] := b[n(n+1)/2, 0, 0, {}];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)

Formula

a(n) = A317166(A000217(n)).

a(n) >= A317273(n).