A278763 - OEIS

%I #6 Dec 03 2016 11:52:35

%S 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,

%T 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,

%U 44,56,55,30,6,1,2,5,9,17,28,46,65,80,70,36

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

%C 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.

%H Clark Kimberling, <a href="/A278763/b278763.txt">Table of n, a(n) for n = 1..1000</a>

%e First 9 rows (for n = 2 to 10):

%e   1;

%e   1,  1;

%e   1,  2,  2;

%e   1,  2,  4,  2;

%e   1,  2,  5,  6,  3;

%e   1,  2,  5,  8,  9,  3;

%e   1,  2,  5,  9, 14, 12,  4;

%e   1,  2,  5,  9, 16, 20, 16,  4;

%e   1,  2,  5,  9, 17, 25, 30, 20,  5;

%e   1,  2,  5,  9, 17, 27, 39, 40, 25,  5;

%e (See also the Example at A278762, for n = 5.)

%t p[n_] := p[n] = IntegerPartitions[n];

%t s[n_, k_] := s[n, k] = Select[p[n], Length[#] == k &];

%t x[n_, k_] := x[n, k] = Map[Length, Map[Union, s[n, k]]];

%t b[h_] := b[h] = h (h - 1)/2;

%t e[n_, k_] := e[n, k] = Total[Map[b, x[n, k]]];

%t Flatten[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]]  (* A278762 sequence *)

%t TableForm[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]]  (* A278762 triangle *)

%t Flatten[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]]  (* A278763 sequence *)

%t TableForm[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]]  (* A278763 triangle *)

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

%K nonn,easy

%O 1,5

%A _Clark Kimberling_, Nov 30 2016