

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.


1



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
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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: A071435 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



