A278763 Triangular array: row n shows the number of edges in successive levels of a graph of the partitions of n; see Comments.

1, 1, 1, 1, 2, 2, 1, 2, 4, 2, 1, 2, 5, 6, 3, 1, 2, 5, 8, 9, 3, 1, 2, 5, 9, 14, 12, 4, 1, 2, 5, 9, 16, 20, 16, 4, 1, 2, 5, 9, 17, 25, 30, 20, 5, 1, 2, 5, 9, 17, 27, 39, 40, 25, 5, 1, 2, 5, 9, 17, 28, 44, 56, 55, 30, 6, 1, 2, 5, 9, 17, 28, 46, 65, 80, 70, 36

Offset

1,5

Comments

The k-th number in row n (with rows numbered 2,3,4,...) is the number of edges from partitions of n into k parts to partitions of n into k-1 parts, for k = n..2, where partitions p and q share an edge if p has one more part than q, and exactly one part of p is a sum of two parts of q. The limiting row is A000097, which also gives the row sums.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

First 9 rows (for n = 2 to 10):
  1;
  1,  1;
  1,  2,  2;
  1,  2,  4,  2;
  1,  2,  5,  6,  3;
  1,  2,  5,  8,  9,  3;
  1,  2,  5,  9, 14, 12,  4;
  1,  2,  5,  9, 16, 20, 16,  4;
  1,  2,  5,  9, 17, 25, 30, 20,  5;
  1,  2,  5,  9, 17, 27, 39, 40, 25,  5;
(See also the Example at A278762, for n = 5.)

Mathematica

p[n_] := p[n] = IntegerPartitions[n];
s[n_, k_] := s[n, k] = Select[p[n], Length[#] == k &];
x[n_, k_] := x[n, k] = Map[Length, Map[Union, s[n, k]]];
b[h_] := b[h] = h (h - 1)/2;
e[n_, k_] := e[n, k] = Total[Map[b, x[n, k]]];
Flatten[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]]  (* A278762 sequence *)
TableForm[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]]  (* A278762 triangle *)
Flatten[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]]  (* A278763 sequence *)
TableForm[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]]  (* A278763 triangle *)

Crossrefs

Cf. A000041, A000097 (row sums), A278762.

Sequence in context: A156260 A056671 A354349 * A278762 A055076 A069780

Adjacent sequences: A278760 A278761 A278762 * A278764 A278765 A278766

Keyword

nonn,easy

Author

Clark Kimberling, Nov 30 2016

Status

approved