OFFSET
0,2
COMMENTS
Let S be a fixed matching of size n in a complete graph G with >= 4n vertices. Given T,T' (also matchings of size n), define the equivalence relation where T ~ T' if and only if there exists an automorphism of G that maps edges in T to edges in T' while mapping edges in S to edges in S. Then the number of equivalence classes is a(n).
a(n) is the number of partitions of 2n with 4 kinds of parts (types 1,2,3,4) where (i) all parts of types 1,2 are odd and all parts of types 3,4 are even; and (ii) the number of type 1 and type 2 parts are equal.
LINKS
Yu Hin Au, Nathan Lindzey, and Levent Tunçel, Matchings, hypergraphs, association schemes, and semidefinite optimization, arXiv:2008.08628 [math.CO], 2020.
EXAMPLE
To see a(2)=10, let S = {{1,2},{3,4}}. Then a representative from each of the 10 equivalence classes are
1. {{1,2}, {3,4}}
2. {{1,3}, {2,4}}
3. {{1,5}, {3,4}}
4. {{1,3}, {4,5}}
5. {{1,2}, {5,6}}
6. {{1,3}, {5,6}}
7. {{1,5}, {2,6}}
8. {{1,5}, {3,6}}
9. {{1,5}, {6,7}}
10. {{5,6}, {7,8}}
CROSSREFS
If the equivalence relation is defined as T~T' if and only if there exists an automorphism of G mapping union of S,T to union of S,T' (i.e., the map does not necessarily fix edges in S), then we obtain A305168.
KEYWORD
nonn
AUTHOR
Yu Hin Au, Aug 31 2018
STATUS
approved