%N The number of idempotents in the Jones (or Temperley-Lieb) monoid on the set [1..n].
%C The Jones monoid is the set of partitions on [1..2n] with classes of size 2, which can be drawn as a planar graph, and multiplication inherited from the Brauer monoid, which contains the Jones monoid as a subsemigroup. The multiplication is defined in Halverson and Ram.
%C These numbers were produced using the Semigroups (2.0) package for GAP 4.7.
%C No general formula is known for the number of idempotents in the Jones monoid.
%H Attila Egri-Nagy, Nick Loughlin, and James Mitchell <a href="/A225798/b225798.txt">Table of n, a(n) for n = 1..30</a> (a(1) to a(21) from Attila Egri-Nagy, a(22)-a(24) from Nick Loughlin, a(25)-a(30) from James Mitchell)
%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 I. Dolinka, J. East et al, <a href="http://arxiv.org/abs/1507.04838">Idempotent Statistics of the Motzkin and Jones Monoids</a>, arXiv:1507.04838 [math.CO], 2015. Table 4 and 5.
%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 [1..18] do
%o Print(NrIdempotents(JonesMonoid(i)), "\n");
%Y Cf. A000108, A227545, A225797.
%A _James Mitchell_, Jul 27 2013
%E a(20)-a(21) from _Attila Egri-Nagy_, Sep 12 2014
%E a(22)-a(24) from _Nick Loughlin_, Jan 23 2015
%E a(25)-a(30) from _James Mitchell_, May 21 2016