A371828 Number of labeled n-vertex hypergraphs (or set systems) that have a solution to the One Up puzzle.

1, 2, 7, 54, 993, 48868

Offset

0,2

Comments

Here, a hypergraph is a set of nonempty subsets (hyperedges) of the set of vertices.

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.

Links

Table of n, a(n) for n=0..5.

Example

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.
         n = 0     n = 1       n = 2           n = 3
  ---------------------------------------------------
         {} (1)    {} (1)      {} (1)          {} (1)
                  {1} (1)     {1} (2)         {1} (3)
                             {12} (1)        {12} (3)
                            {1,2} (1)       {123} (1)
                           {1,12} (2)       {1,2} (3)
                                           {1,12} (6)
                                           {1,23} (3)
                                          {1,123} (3)
                                          {12,13} (3)
                                         {12,123} (3)
                                          {1,2,3} (1)
                                         {1,2,13} (6)
                                        {1,12,13} (3)
                                        {1,12,23} (6)
                                       {1,12,123} (6)
                                      {1,2,13,23} (3)
  ---------------------------------------------------
  a(n):      1         2           7              54

Crossrefs

Cf. A371476, A371829 (unlabeled hypergraphs).

Sequence in context: A283335 A326207 A280221 * A227381 A182055 A358944

Adjacent sequences: A371825 A371826 A371827 * A371829 A371830 A371831

Keyword

nonn,more

Author

Pontus von Brömssen, Apr 07 2024

Status

approved