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%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
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