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

Offset: 3

Author

Gordon F. Royle, Mar 18 2004

Keywords

• nonn
• tabf

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.

Examples

			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;
		

Links

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

Crossrefs

Cf. A000665, A051240.

Programs

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] /.
Table[s[i] -> 1 + x^i, {i, 1, Binomial[n, 3]}], x], {n,3,7}] // Grid (* Geoffrey Critzer, Oct 28 2015 *)