OFFSET
1,2
COMMENTS
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.
Number of hypergraphs of order n that admit a unique perfect fractional matching assigning a positive weight to each hyperedge.
LINKS
Andrew Howroyd, PARI Program
Dylan Laplace Mermoud, Michel Grabisch and Peter Sudhölter, Minimal balanced collections: generation, applications and generalization, Documents de travail du Centre d'Économie de la Sorbonne, 2023.
Dylan Laplace Mermoud and Pierre Popoli, Combinatorics on Social Configurations, arXiv:2406.16409 [cs.GT], 2024. See p. 2.
Lloyd S. Shapley, On balanced sets and cores, Rand Corporation Research Memoranda Series, RM-4601-PR, 1965.
Lloyd S. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly, (1967)(14)4, 453-460.
EXAMPLE
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.
From Andrew Howroyd, Jan 14 2023: (Start)
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:
a b c d
{abc} 2/3 2/3 2/3 -
{ad} 1/3 - - 1/3
{bd} - 1/3 - 1/3
{cd} - - 1/3 1/3
In the table it can be seen that the values in each row are equal while those in each column sum to 1.
(End)
PROG
(PARI) \\ See link. - Andrew Howroyd, Jan 14 2023
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Dylan Laplace Mermoud, Jun 16 2022
STATUS
approved