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.

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, 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, 2, 5, 4, 3, 1, 2, 4, 1, 2, 6, 5, 1, 2, 8, 3, 2, 1, 1

Offset: 3

Author

Clark Kimberling, Apr 30 2014

Keywords

• nonn
• tabf
• easy

Comments

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)).

Examples

			Rows 3 to 15 (counting the top row as row 3):
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 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
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.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 3..3000

Crossrefs

Cf. A000009, A108796.

Programs

Mathematica

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 *)