A370945 -id:A370945 - cp's OEIS frontend

A370946 Number of partitions of [n] whose non-singleton elements sum to n.

1, 0, 0, 1, 1, 2, 3, 4, 5, 7, 12, 14, 20, 26, 36, 54, 68, 90, 120, 157, 202, 296, 360, 480, 612, 803, 1006, 1317, 1764, 2198, 2821, 3592, 4552, 5754, 7269, 9074, 11990, 14646, 18586, 23112, 29208, 35972, 45277, 55584, 69350, 87881, 107609, 133068, 165038

Offset: 0

Alois P. Heinz, Mar 06 2024

nonn

			a(0) = 1: the empty partition.
a(3) = 1: 12|3.
a(4) = 1: 13|2|4.
a(5) = 2: 1|23|4|5, 14|2|3|5.
a(6) = 3: 123|4|5|6, 1|24|3|5|6, 15|2|3|4|6.
a(7) = 4: 124|3|5|6|7, 1|2|34|5|6|7, 1|25|3|4|6|7, 16|2|3|4|5|7.
a(8) = 5: 125|3|4|6|7|8, 134|2|5|6|7|8, 1|2|35|4|6|7|8, 1|26|3|4|5|7|8, 17|2|3|4|5|6|8.
a(9) = 7: 126|3|4|5|7|8|9, 135|2|4|6|7|8|9, 1|234|5|6|7|8|9, 1|2|3|45|6|7|8|9, 1|2|36|4|5|7|8|9, 1|27|3|4|5|6|8|9, 18|2|3|4|5|6|7|9.
a(10) = 12: 1234|5|6|7|8|9|10, 12|34|5|6|7|8|9|10, 127|3|4|5|6|8|9|10, 13|24|5|6|7|8|9|10, 136|2|4|5|7|8|9|10, 14|23|5|6|7|8|9|10, 1|235|4|6|7|8|9|10, 145|2|3|6|7|8|9|10, 1|2|3|46|5|7|8|9|10, 1|2|37|4|5|6|8|9|10, 1|28|3|4|5|6|7|9|10, 19|2|3|4|5|6|7|8|10.

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

Cf. A161680, A370945.

h:= proc(n) option remember; `if`(n=0, 1,
      add(h(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, h(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..48);

h[n_] := h[n] = If[n == 0, 1, Sum[h[n-j]*Binomial[n-1, j-1], {j, 2, n}]];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0, If[n == 0, h[m], b[n, i - 1, m] + b[n - i, Min[n - i, i - 1], m + 1]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Mar 08 2024, after Alois P. Heinz *)

a(n) = A370945(n,n*(n-1)/2).

A370947 Number of partitions of [n] whose singletons sum to n.

Original entry on oeis.org

1, 1, 0, 1, 2, 6, 20, 78, 307, 1486, 6974, 38584, 212268, 1321886, 8186322, 57015161, 391153290, 2976480926, 22534577137, 185638964675, 1522358748758, 13558705354828, 119620910388056, 1137343427864934, 10770667246889494, 108819371313460263, 1095389086585963202

Offset: 0

Alois P. Heinz, Mar 06 2024

nonn

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(3) = 1: 12|3.
a(4) = 2: 123|4, 1|24|3.
a(5) = 6: 1234|5, 12|34|5, 13|24|5, 14|23|5, 1|235|4, 145|2|3.
a(6) = 20: 12345|6, 123|45|6, 124|35|6, 125|34|6, 12|345|6, 134|25|6, 135|24|6, 13|245|6, 1356|2|4, 13|2|4|56, 145|23|6, 14|235|6, 15|234|6, 1|2346|5, 1|23|46|5, 1|24|36|5, 1|26|34|5, 15|2|36|4, 16|2|35|4, 1|2|3|456.

Alois P. Heinz, Table of n, a(n) for n = 0..577
Wikipedia, Partition of a set

Main diagonal of A370945.

Cf. A000009, A000110, A000296.

h:= proc(n) option remember; `if`(n=0, 1,
      add(h(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, h(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..26);

h[n_] := h[n] = If[n == 0, 1, Sum[h[n-j]*Binomial[n-1, j-1], {j, 2, n}]];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0, If[n == 0, h[m], b[n, i - 1, m] + b[n - i, Min[n - i, i - 1], m - 1]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 08 2024, after Alois P. Heinz *)

a(n) = A370945(n,n).

cp's OEIS Frontend

A370946 Number of partitions of [n] whose non-singleton elements sum to n.

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