A225797 The number of idempotents in the partition monoid on [1..n].

2, 12, 114, 1512, 25826, 541254, 13479500, 389855014, 12870896154, 478623817564, 19835696733562, 908279560428462, 45625913238986060, 2499342642591607902, 148545280714724993650, 9537237096314268691724

Offset: 1

Author

James Mitchell, Jul 27 2013

Keywords

• nonn

Comments

The partition monoid is the set of partitions on [1..2n] and multiplication as defined in Halverson and Ram.

No general formula is known for the number of idempotents in the partition monoid.

a(2) to a(8) were first produced using the Semigroups package for GAP, which contains code based on earlier calculations by Max Neunhoeffer.

Links

• James Mitchell, Table of n, a(n) for n = 1..115
• I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
• T. Halverson, A. Ram, Partition algebras, European J. Combin. 26 (6) (2005) 869-921.
• J. D. Mitchell et al., Semigroups package for GAP.

Crossrefs

Cf. A227545.

Programs

GAP

for i in [2 .. 8] do
  Print(NrIdempotents(PartitionMonoid(i)), "\n");
od;

Extensions

a(9)-a(12) from James East, Feb 07 2014

a(13) onwards from James Mitchell, May 23 2016