A368204 Number of partitions of [n] whose block minima sum to n.

1, 1, 0, 2, 2, 2, 29, 56, 191, 380, 5097, 14288, 74359, 283884, 1106529, 13588409, 53640963, 350573155, 1867738775, 10770352150, 50050737949, 744605446778, 3615378756421, 29368052533243, 195027586980839, 1442227919200245, 8964685271444243, 61478734886319324

Offset: 0

Author

Alois P. Heinz, Dec 16 2023

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 0.
a(3) = 2: 13|2, 1|23.
a(4) = 2: 124|3, 12|34.
a(5) = 2: 1235|4, 123|45.
a(6) = 29: 12346|5, 1234|56, 1456|2|3, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|36, 146|2|35, 14|26|35, 14|2|356, 156|24|3, 15|246|3, 15|24|36, 16|245|3, 1|2456|3, 1|245|36, 16|24|35, 1|246|35, 1|24|356, 156|2|34, 15|26|34, 15|2|346, 16|25|34, 1|256|34, 1|25|346, 16|2|345, 1|26|345, 1|2|3456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..677
• Wikipedia, Partition of a set

Crossrefs

Main diagonal of A124327.

Cf. A365441, A368246.

Programs

Maple

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 0, `if`(t=0, 0,
      t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
    end:
a:= n-> b(n, 1, 0, n):
seq(a(n), n=0..42);

Mathematica

b[n_, i_, t_, m_] := b[n, i, t, m] = If[n == 0, t^(m - i + 1),
   If[(i + m)*(m + 1 - i)/2 < n || i > n, 0, If[t == 0, 0,
   t*b[n, i + 1, t, m]] + b[n - i, i + 1, t + 1, m]]];
a[n_] := If[n == 0, 1, b[n, 1, 0, n]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jun 10 2024, after Alois P. Heinz *)

Formula

a(n) = A124327(n,n).