%I #59 Jun 28 2024 11:22:49
%S 1,2,6,42,1292,200214,132422036
%N a(n) is the number of minimal balanced collections on a ground set of n labeled elements.
%C A balanced collection, a.k.a. a balanced set, is a collection of subsets of a ground set N together with a system of weights such that, for any element i of N, the sum of the weights associated with the subsets containing i is one. A minimal balanced collection is a minimal (w.r.t. inclusion) element among the balanced collections.
%C Number of hypergraphs of order n that admit a unique perfect fractional matching assigning a positive weight to each hyperedge.
%H Andrew Howroyd, <a href="/A355042/a355042.txt">PARI Program</a>
%H Dylan Laplace Mermoud, Michel Grabisch and Peter Sudhölter, <a href="https://ideas.repec.org/p/mse/cesdoc/23001.html">Minimal balanced collections: generation, applications and generalization</a>, Documents de travail du Centre d'Économie de la Sorbonne, 2023.
%H Dylan Laplace Mermoud and Pierre Popoli, <a href="https://arxiv.org/abs/2406.16409">Combinatorics on Social Configurations</a>, arXiv:2406.16409 [cs.GT], 2024. See p. 2.
%H Lloyd S. Shapley, <a href="https://www.rand.org/pubs/research_memoranda/RM4601.html">On balanced sets and cores</a>, Rand Corporation Research Memoranda Series, RM-4601-PR, 1965.
%H Lloyd S. Shapley, <a href="https://doi.org/10.1002/nav.3800140404">On balanced sets and cores</a>, Naval Research Logistics Quarterly, (1967)(14)4, 453-460.
%e The a(3) = 6 solutions are {{a}, {b}, {c}}, {{a}, {bc}}, {{b}, {ac}}, {{c}, {ab}}, {{abc}}, {{ab}, {bc}, {ac}}. The collection {{a}, {b}, {c}, {abc}} is not a solution because it is the union of two other solutions and is therefore not minimal.
%e From _Andrew Howroyd_, Jan 14 2023: (Start)
%e One solution for n = 4 is {{abc}, {ad}, {bd}, {cd}}. In this solution the set {abc} has weight 2/3 while the others have weight 1/3. This solution is shown in tabular form below:
%e a b c d
%e {abc} 2/3 2/3 2/3 -
%e {ad} 1/3 - - 1/3
%e {bd} - 1/3 - 1/3
%e {cd} - - 1/3 1/3
%e In the table it can be seen that the values in each row are equal while those in each column sum to 1.
%e (End)
%o (PARI) \\ See link. - _Andrew Howroyd_, Jan 14 2023
%Y Cf. A000110 (number of minimal balanced collections with a system of weights containing only the weight one).
%K nonn,hard,more
%O 1,2
%A _Dylan Laplace Mermoud_, Jun 16 2022