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

0,3

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

Table of n, a(n) for n=0..22.

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

Cf. A023997, A225797, A277379.

Sequence in context: A318694 A281433 A277379 * A127113 A051540 A272135

Adjacent sequences: A227542 A227543 A227544 * A227546 A227547 A227548

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