cp's OEIS Frontend

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).

%I A092337 #15 Nov 01 2015 10:23:00
%S A092337 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,
%T A092337 312,249,161,94,43,21,7,3,1,1,1,1,3,10,38,137,509,1760,5557,15709,
%U A092337 39433,87659,172933,303277,473827,660950,824410,920446,920446,824410,660950
%N 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).
%C A092337 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.
%H A092337 Edgar M. Palmer, <a href="http://dx.doi.org/10.1016/0012-365X(73)90069-1">On the number of n-plexes</a>, Discrete Math., 6 (1973), 377-390.
%e A092337 Triangle T(n,m) begins:
%e A092337 1, 1;
%e A092337 1, 1, 1, 1,  1;
%e A092337 1, 1, 2, 4,  6,  6,  6,   4,   2,   1,   1;
%e A092337 1, 1, 3, 7, 21, 43, 94, 161, 249, 312, 352, 312, 249, 161, 94, 43, 21, 7, 3, 1, 1;
%t A092337 Needs["Combinatorica`"]; Table[A = Subsets[Range[n], {3}];
%t A092337   CoefficientList[CycleIndex[Replace[Map[Sort,System`PermutationReplace[A, SymmetricGroup[n]], {2}],Table[A[[i]] -> i, {i, 1, Length[A]}], 2], s] /.
%t A092337 Table[s[i] -> 1 + x^i, {i, 1, Binomial[n, 3]}], x], {n,3,7}] // Grid (* _Geoffrey Critzer_, Oct 28 2015 *)
%Y A092337 Cf. A000665, A051240.
%K A092337 nonn,tabf
%O A092337 3,10
%A A092337 _Gordon F. Royle_, Mar 18 2004