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A278762 Triangular array: row n shows the number of edges in successive levels of a graph of the partitions of n; see Comments. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A156260 A056671 A278763 * A055076 A069780 A210531

Adjacent sequences:  A278759 A278760 A278761 * A278763 A278764 A278765

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 30 2016

STATUS

approved

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Last modified November 15 15:59 EST 2018. Contains 317239 sequences. (Running on oeis4.)