A317329 Number of permutations of [n] with equal lengths of increasing runs.

1, 2, 2, 7, 2, 82, 2, 1456, 1515, 50774, 2, 3052874, 2, 199364414, 136835794, 19451901825, 2, 2510158074714, 2, 370671075758054, 132705620239756, 69348874393843334, 2, 15772160279898993782, 613498040952503, 4087072509293134292962, 705927677748508225534

Offset: 1

Author

Alois P. Heinz, Jul 25 2018

Keywords

• nonn

Examples

			a(4) = 7: 1234, 1324, 1423, 2314, 2413, 3412, 4321.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..400

Crossrefs

Column k=1 of A317327.

Cf. A000040, A005225.

Programs

Maple

b:= proc(u, o, t, d) option remember; `if`(u+o=0, 1,
      `if`(t=d, add(b(u-j, o+j-1, 1, d), j=1..u),
       add(b(u+j-1, o-j, t+1, d), j=1..o)))
    end:
a:= proc(n) option remember; `if`(n=1, 1, 2+add(
      b(n, 0, d$2), d=numtheory[divisors](n) minus {1, n}))
    end:
seq(a(n), n=1..35);

Mathematica

b[u_, o_, t_, d_] := b[u, o, t, d] = If[u + o == 0, 1,
     If[t == d, Sum[b[u - j, o + j - 1, 1, d], {j, 1, u}],
     Sum[b[u + j - 1, o - j, t + 1, d], {j, 1, o}]]];
a[n_] := a[n] = If[n == 1, 1, 2 + Sum[b[n, 0, d, d], {d, Divisors[n] ~Complement~ {1, n}}]];
Array[a, 35] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

Formula

a(n) = 2 <=> n in { A000040 }.