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A241815 Irregular triangular array : t(n, k) = number of vertices of degree k in graph S(n) of strict partitions, where two partitions have an edge if and only if their intersection is empty.

%I A241815 #7 Jun 18 2015 04:00:43
%S A241815 2,2,3,1,2,1,1,3,1,1,1,2,1,1,3,1,3,1,1,2,2,2,2,1,1,3,2,4,1,1,1,2,4,2,
%T A241815 1,4,1,3,4,4,1,4,1,1,1,3,1,5,1,4,1,2,3,1,1,6,5,6,1,1,2,2,2,1,1,2,6,1,
%U A241815 2,5,4,3,1,2,4,1,2,6,5,1,2,8,3,2,1,1
%N A241815 Irregular triangular array :  t(n, k) = number of vertices of degree k in graph S(n) of strict partitions, where two partitions have an edge if and only if their intersection is empty.
%C A241815 The strict partitions of n are the partitions of n that have distinct parts, as in A000009 (which gives the row sums of t(n,k)).
%H A241815 Clark Kimberling, <a href="/A241815/b241815.txt">Table of n, a(n) for n = 3..3000</a>
%e A241815 Rows 3 to 15 (counting the top row as row 3):
%e A241815 2
%e A241815 2
%e A241815 3
%e A241815 1 2 1
%e A241815 1 3 1
%e A241815 1 1 2 1 1
%e A241815 3 1 3 1
%e A241815 1 2 2 2 2 1
%e A241815 1 3 2 4 1 1
%e A241815 1 2 4 2 1 4 1
%e A241815 3 4 4 1 4 1 1
%e A241815 1 3 1 5 1 4 1 2 3 1
%e A241815 1 6 5 6 1 1 2 2 2 1
%e A241815 The graph S(3) is given by 3 -> 21; S(4), by 4 -> 31, S(6), by 6 -> 51, 6 -> 42, 6 -> 321, 51 -> 42, and S(8), by 8 -> 71, 8 -> 62, 8 -> 53, 8 -> 521, 8 -> 431, 71 -> 62, 71 -> 53, 62, -> 53, 62 -> 431.  The vertices of S(8) and their degrees d are easily read from the graph:  d(521) = 1, d(431) = 2, d(71) = 3, d(53) = 3, d(62) = 4, and d(8) = 5, so that row 8 (counting the top row as row 3) is 1,1,2,1,1.
%t A241815 z = 20; p[n_] := p[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; d[n_] :=  Table[Map[{p[n][[k]], #} &, DeleteCases[Flatten[Select[Map[{#, Intersection[p[n][[k]], #]} &, p[n]], #[[2]] == {} &], 1], {}]], {k, Length[p[n]]}]; u[n_] := Sort[Split[Sort[Flatten[d[n], 2]]]]; t = Table[Map[Length, u[n]]/2, {n, 1, z}]; Join[{0, 0}, Flatten[t]]  (* _Peter J. C. Moses_, Apr 17 2014 *)
%Y A241815 Cf. A000009, A108796.
%K A241815 nonn,tabf,easy
%O A241815 3,1
%A A241815 _Clark Kimberling_, Apr 30 2014