A278762 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, 2, 2, 1, 2, 4, 2, 1, 3, 6, 5, 2, 1, 3, 9, 8, 5, 2, 1, 4, 12, 14, 9, 5, 2, 1, 4, 16, 20, 16, 9, 5, 2, 1, 5, 20, 30, 25, 17, 9, 5, 2, 1, 5, 25, 40, 39, 27, 17, 9, 5, 2, 1, 6, 30, 55, 56, 44, 28, 17, 9, 5, 2, 1, 6, 36, 70, 80, 65, 46, 28, 17, 9, 5, 2

Offset

1,4

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 = 1..n-1, where partitions p and q share an edge if q has one more part than p, and exactly one part of q is a sum of two parts of p.

Links

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

Example

First 9 rows (for n = 2 to 10):
  1;
  1,  1;
  2,  2,  1;
  2,  4,  2,  1;
  3,  6,  5,  2,  1;
  3,  9,  8,  5,  2,  1;
  4, 12, 14,  9,  5,  2,  1;
  4, 16, 20, 16,  9,  5,  2,  1;
  5, 20, 30, 25, 17,  9,  5,  2,  1;
The 7 partitions of 5 are arranged as shown below, with edges 5-to-41, 5-to-32, 41-to-311, 41-to-221, 32-to-311, 32-to-221, 311-to-2111, 221-to-2111, and 2111-to-11111.
        41     311
5                             2111     11111
        32      221
For row 5, there are 2 edges from 5 to {41, 32}, 4 edges from {41,32} to {311,221}, 2 edges from {311,221} to 2111, and 1 edge from 2111 to 11111; consequently, row 5 is 2 4 2 1.

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), A278763.

Sequence in context: A056671 A354349 A278763 * A055076 A069780 A369307

Adjacent sequences: A278759 A278760 A278761 * A278763 A278764 A278765

Keyword

nonn,easy

Author

Clark Kimberling, Nov 30 2016

Status

approved