A317273 Number of permutations of [n*(n+1)/2] whose lengths of increasing runs are the positive integers from 1 to n.

1, 1, 4, 202, 163692, 2487100956, 832252747110528, 7116720347983770858600, 1776529280247277318394451118272, 14580103976468323893693256154922439405632, 4377460729080839690885111988468699720430287682744896, 52959485251272238069446517666752040946228209263610778166878160384

Offset: 0

Author

Alois P. Heinz, Jul 25 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..14

Crossrefs

Cf. A000027, A000217, A317165.

Programs

Maple

g:= (n, s)-> `if`(n in s, 1, 0):
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 minus {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, {$0..n}):
seq(a(n), n=0..10);

Mathematica

g[n_, s_] := If[MemberQ[s, n], 1, 0];
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 ~Complement~ {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, Range[0, n]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)

Formula

a(n) = A317327(A000217(n),n).

a(n) <= A317165(n).