A227545 The number of idempotents in the Brauer monoid on [1..n].

1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016, 38350749144768938770432

Offset: 0

Author

James Mitchell, Jul 15 2013

Keywords

• nonn

Comments

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.

These numbers were produced using the Semigroups (2.0) package for GAP 4.7.

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

Links

• 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.

Crossrefs

Cf. A023997, A225797, A277379.

Programs

GAP

for i in [1..11] do
  Print(NrIdempotents(BrauerMonoid(i)), "\n");
od;

Mathematica

nn = 44; ee = Table[0, nn+1]; ee[[1]] = 1;
e[n_] := e[n] = ee[[n+1]];
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}]
];
ee (* Jean-François Alcover, Jul 21 2018, after Joerg Arndt *)

PARI

N=44; E=vector(N+1); E[1]=1;
e(n)=E[n+1];
{ for (n=1, N,
E[n+1]=
     sum(i=1,n\2,binomial(n-1,2*i-1)*(2*i-1)!*e(n-2*i)) +
     sum(i=0,(n-1)\2,binomial(n-1,2*i)*(2*i+1)!*e(n-2*i-1))
); }
print(E);
\\ Joerg Arndt, Oct 12 2016

Extensions

Terms a(13)-a(17) from James East, Dec 23 2013

More terms from Joerg Arndt, Oct 12 2016