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%I #54 Jul 21 2018 10:05:13
%S 1,1,2,10,40,296,1936,17872,164480,1820800,21442816,279255296,
%T 3967316992,59837670400,988024924160,17009993230336,318566665977856,
%U 6177885274406912,129053377688043520,2786107670662021120,64136976817284448256,1525720008470138454016,38350749144768938770432
%N The number of idempotents in the Brauer monoid on [1..n].
%C The Brauer monoid is the set of partitions on [1..2n] with classes of size 2 and multiplication inherited from the partition monoid, which contains the Brauer monoid as a subsemigroup. The multiplication is defined in Halverson & 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 Brauer monoid.
%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://www.ms.unimelb.edu.au/~ram/Publications/2005EJCv26p869.pdf">Partition algebras</a>, European J. Combin. 26 (6) (2005) 869-921.
%t nn = 44; ee = Table[0, nn+1]; ee[[1]] = 1;
%t e[n_] := e[n] = ee[[n+1]];
%t For[n = 1, n <= nn, n++, ee[[n+1]] = Sum[Binomial[n-1, 2i-1] (2i-1)! e[n-2i], {i, 1, n/2}] + Sum[Binomial[n-1, 2i] (2i+1)! e[n-2i-1], {i, 0, (n-1)/2}]
%t ];
%t ee (* _Jean-François Alcover_, Jul 21 2018, after _Joerg Arndt_ *)
%o (GAP) for i in [1..11] do
%o Print(NrIdempotents(BrauerMonoid(i)), "\n");
%o od;
%o (PARI)
%o N=44; E=vector(N+1); E[1]=1;
%o e(n)=E[n+1];
%o { for (n=1, N,
%o E[n+1]=
%o sum(i=1,n\2,binomial(n-1,2*i-1)*(2*i-1)!*e(n-2*i)) +
%o sum(i=0,(n-1)\2,binomial(n-1,2*i)*(2*i+1)!*e(n-2*i-1))
%o ); }
%o print(E);
%o \\ _Joerg Arndt_, Oct 12 2016
%Y Cf. A023997, A225797, A277379.
%K nonn
%O 0,3
%A _James Mitchell_, Jul 15 2013
%E Terms a(13)-a(17) from _James East_, Dec 23 2013
%E More terms from _Joerg Arndt_, Oct 12 2016
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