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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..677

Wikipedia, Partition of a set

Formula

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

Example

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.

Maple

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2<n or i>n, 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 *)

Crossrefs

Main diagonal of A124327.

Cf. A365441, A368246.

Sequence in context: A157979 A147975 A083148 * A318166 A241845 A254131

Adjacent sequences: A368201 A368202 A368203 * A368205 A368206 A368207

Keyword

nonn

Author

Alois P. Heinz, Dec 16 2023

Status

approved