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 A227545 The number of idempotents in the Brauer monoid on [1..n]. 4
 1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016, 38350749144768938770432 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified January 21 13:07 EST 2019. Contains 319350 sequences. (Running on oeis4.)