A368102 Total number of partitions of [n-s] whose block maxima sum to s, summed over all s.

1, 0, 1, 0, 1, 1, 1, 1, 3, 2, 3, 6, 5, 9, 14, 13, 20, 40, 31, 56, 86, 105, 127, 227, 244, 394, 520, 665, 911, 1536, 1565, 2449, 3507, 4719, 5931, 9061, 11151, 16911, 21774, 29798, 39804, 60411, 71865, 104399, 144999, 197907, 253430, 370044, 478764, 694807

Offset: 0

Author

Alois P. Heinz, Dec 11 2023

Keywords

• nonn

Examples

			a(11) = 6: 123|4, 124|3, 13|24, 14|23, 1|2|34, 1|2345.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• Wikipedia, Partition of a set

Crossrefs

Antidiagonal sums of A367955.

Cf. A365821.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> add(coeff(b(n-k, 0), x, k), k=ceil(n/2)..n):
seq(a(n), n=0..80);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 add(b(k, n-k, 0), k=ceil(n/2)..n):
seq(a(n), n=0..80);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, Sum[b[k, n - k, 0], {k, Ceiling[n/2], n}]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)