A316587 a(n) = [x^(2n)y^n] Product_{i>=1} 1/((1-x^(2i-1)y^i)(1-x^(2i-1)y^(i-1))(1-x^(2i)y^i)^2).

1, 3, 10, 27, 69, 161, 361, 767, 1578, 3134, 6064, 11432, 21105, 38175, 67863, 118658, 204455, 347439, 583063, 966952, 1586231, 2575474, 4141832, 6600731, 10430455, 16349788, 25434178, 39280676, 60250276, 91810915, 139034070, 209294256, 313269591, 466343647

Offset: 0

Author

Yu Hin Au, Aug 31 2018

Keywords

• nonn

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.

Examples

			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}}
		

Links

• Yu Hin Au, Nathan Lindzey, and Levent Tunçel, Matchings, hypergraphs, association schemes, and semidefinite optimization, arXiv:2008.08628 [math.CO], 2020.

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.