%I #29 May 23 2016 17:04:39
%S 2,12,114,1512,25826,541254,13479500,389855014,12870896154,
%T 478623817564,19835696733562,908279560428462,45625913238986060,
%U 2499342642591607902,148545280714724993650,9537237096314268691724
%N The number of idempotents in the partition monoid on [1..n].
%C The partition monoid is the set of partitions on [1..2n] and multiplication as defined in Halverson and Ram.
%C No general formula is known for the number of idempotents in the partition monoid.
%C a(2) to a(8) were first produced using the Semigroups package for GAP, which contains code based on earlier calculations by Max Neunhoeffer.
%H James Mitchell, <a href="/A225797/b225797.txt">Table of n, a(n) for n = 1..115</a>
%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
%H T. Halverson, A. Ram, <a href="http://dx.doi.org/10.1016/j.ejc.2004.06.005">Partition algebras</a>, European J. Combin. 26 (6) (2005) 869-921.
%H J. D. Mitchell et al., <a href="https://gap-packages.github.io/Semigroups/">Semigroups</a> package for GAP.
%o (GAP) for i in [2 .. 8] do
%o Print(NrIdempotents(PartitionMonoid(i)), "\n");
%o od;
%Y Cf. A227545.
%K nonn
%O 1,1
%A _James Mitchell_, Jul 27 2013
%E a(9)-a(12) from _James East_, Feb 07 2014
%E a(13) onwards from _James Mitchell_, May 23 2016