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

1,2

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Column k=1 of A317327.

Cf. A000040, A005225.

Sequence in context: A211780 A014840 A218756 * A235709 A341760 A342453

Adjacent sequences: A317326 A317327 A317328 * A317330 A317331 A317332

Keyword

nonn

Author

Alois P. Heinz, Jul 25 2018

Status

approved