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%I A371828 #7 Apr 10 2024 09:26:38
%S A371828 1,2,7,54,993,48868
%N A371828 Number of labeled n-vertex hypergraphs (or set systems) that have a solution to the One Up puzzle.
%C A371828 Here, a hypergraph is a set of nonempty subsets (hyperedges) of the set of vertices.
%C A371828 The One Up puzzle on a polyomino is defined in A371476. On a hypergraph, the objective of the puzzle is to assign a positive integer to each vertex in such a way that the vertices of each hyperedge are assigned consecutive numbers starting at 1. In other words, the vertex of a hyperedge of size 1 must be assigned the number 1, the vertices of a hyperedge of size 2 must be assigned the numbers 1 and 2, etc.
%e A371828 The following hypergraphs have solutions to the One Up puzzle. Only one such hypergraph for each isomorphism class is given, with the size of the isomorphism class in parentheses.
%e A371828 n = 0 n = 1 n = 2 n = 3
%e A371828 ---------------------------------------------------
%e A371828 {} (1) {} (1) {} (1) {} (1)
%e A371828 {1} (1) {1} (2) {1} (3)
%e A371828 {12} (1) {12} (3)
%e A371828 {1,2} (1) {123} (1)
%e A371828 {1,12} (2) {1,2} (3)
%e A371828 {1,12} (6)
%e A371828 {1,23} (3)
%e A371828 {1,123} (3)
%e A371828 {12,13} (3)
%e A371828 {12,123} (3)
%e A371828 {1,2,3} (1)
%e A371828 {1,2,13} (6)
%e A371828 {1,12,13} (3)
%e A371828 {1,12,23} (6)
%e A371828 {1,12,123} (6)
%e A371828 {1,2,13,23} (3)
%e A371828 ---------------------------------------------------
%e A371828 a(n): 1 2 7 54
%Y A371828 Cf. A371476, A371829 (unlabeled hypergraphs).
%K A371828 nonn,more
%O A371828 0,2
%A A371828 _Pontus von Brömssen_, Apr 07 2024