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

3,1

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)).

Links

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

Example

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.

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 *)

Crossrefs

Cf. A000009, A108796.

Sequence in context: A365484 A335814 A119428 * A051521 A319562 A171810

Adjacent sequences: A241812 A241813 A241814 * A241816 A241817 A241818

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved