A092337 - OEIS

A092337

Triangle read by rows: T(n,m) = number of 3-uniform hypergraphs with m hyperedges on n unlabeled nodes, where 0 <= m <= C(n,3).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1, 1, 1, 3, 7, 21, 43, 94, 161, 249, 312, 352, 312, 249, 161, 94, 43, 21, 7, 3, 1, 1, 1, 1, 3, 10, 38, 137, 509, 1760, 5557, 15709, 39433, 87659, 172933, 303277, 473827, 660950, 824410, 920446, 920446, 824410, 660950

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OFFSET

3,10

COMMENTS

A 3-uniform hypergraph is a set of 3-subsets of the nodes with isomorphism determined by permutations of the node set. The numbers are calculated using Polya enumeration from the cycle index of the symmetric group Sym(n) in its action on 3-subsets of an n-set, which can easily be computed by MAGMA or GAP. A000665 is the sum of each row of the triangle.

LINKS

Table of n, a(n) for n=3..62.

Edgar M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390.

EXAMPLE

Triangle T(n,m) begins:

1, 1;

1, 1, 1, 1, 1;

1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1;

1, 1, 3, 7, 21, 43, 94, 161, 249, 312, 352, 312, 249, 161, 94, 43, 21, 7, 3, 1, 1;

MATHEMATICA

Needs["Combinatorica`"]; Table[A = Subsets[Range[n], {3}];

CoefficientList[CycleIndex[Replace[Map[Sort, System`PermutationReplace[A, SymmetricGroup[n]], {2}], Table[A[[i]] -> i, {i, 1, Length[A]}], 2], s] /.