OFFSET
0,3
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.
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 *)
PROG
(GAP) for i in [1..11] do
Print(NrIdempotents(BrauerMonoid(i)), "\n");
od;
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
James Mitchell, Jul 15 2013
EXTENSIONS
Terms a(13)-a(17) from James East, Dec 23 2013
More terms from Joerg Arndt, Oct 12 2016
STATUS
approved