A322875 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals two.

0, 1, 5, 21, 86, 361, 1584, 7315, 35635, 183080, 990659, 5635021, 33622161, 209973099, 1369560267, 9310957518, 65852852210, 483672626464, 3683088047043, 29033382412670, 236591717703447, 1990467019391404, 17268021545339042, 154304401318961489

Offset: 2

Author

Alois P. Heinz, Dec 29 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..590
• Wikipedia, Partition of a set

Crossrefs

Column k=2 of A287215.

Cf. A026898, A287252.

Programs

Maple

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
a:= n-> (k-> A(n, k)-A(n, k-1))(2):
seq(a(n), n=2..30);

Mathematica

b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m b[n - 1, k, m, l]];
A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
a[n_] := With[{k = 2}, A[n, k] - A[n, k - 1]];
a /@ Range[2, 30] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

a(n) = A287252(n) - A026898(n-1).