A225797 -id:A225797 - cp's OEIS frontend

A225798 The number of idempotents in the Jones (or Temperley-Lieb) monoid on the set [1..n].

1, 2, 5, 12, 36, 96, 311, 886, 3000, 8944, 31192, 96138, 342562, 1083028, 3923351, 12656024, 46455770, 152325850, 565212506, 1878551444, 7033866580, 23645970022, 89222991344, 302879546290, 1150480017950, 3938480377496, 15047312553918, 51892071842570, 199274492098480, 691680497233180

Offset: 1

James Mitchell, Jul 27 2013

nonn

The Jones monoid is the set of partitions on [1..2n] with classes of size 2, which can be drawn as a planar graph, and multiplication inherited from the Brauer monoid, which contains the Jones monoid as a subsemigroup. The multiplication is defined in Halverson and 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 Jones monoid.

Attila Egri-Nagy, Nick Loughlin, and James Mitchell Table of n, a(n) for n = 1..30 (a(1) to a(21) from Attila Egri-Nagy, a(22)-a(24) from Nick Loughlin, a(25)-a(30) from James Mitchell)
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.
I. Dolinka, J. East et al, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015. Table 4 and 5.
T. Halverson, A. Ram, Partition algebras, European J. Combin. 26 (6) (2005) 869-921.
J. D. Mitchell et al., Semigroups package for GAP.

Cf. A000108, A227545, A225797.

for i in [1..18] do
Print(NrIdempotents(JonesMonoid(i)), "\n");
od;

a(20)-a(21) from Attila Egri-Nagy, Sep 12 2014
a(22)-a(24) from Nick Loughlin, Jan 23 2015
a(25)-a(30) from James Mitchell, May 21 2016

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

Original entry on oeis.org

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

James Mitchell, Jul 15 2013

nonn

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.

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.

Cf. A023997, A225797, A277379.

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

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 *)

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

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

More terms from Joerg Arndt, Oct 12 2016

A286867 Number of idempotents in the twisted planar partition monoid PP_n^tau.

Original entry on oeis.org

1, 1, 6, 44, 362, 3226, 30488, 301460, 3090020, 32618046, 345515557

Offset: 0

James East, Oct 05 2017

Values were computed using the Semigroups package for GAP.

Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Cf. A225797, A289620.

A256035 Number of idempotent basis elements in partition monoid P_n.

Original entry on oeis.org

1, 1, 6, 59, 807, 14102, 301039, 7618613, 223586932, 7482796089, 281882090283

Offset: 0

N. J. A. Sloane, Mar 14 2015

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. See Table 3.

Cf. A060639, A256033, A256034, A225797.

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A225798 The number of idempotents in the Jones (or Temperley-Lieb) monoid on the set [1..n].

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

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A286867 Number of idempotents in the twisted planar partition monoid PP_n^tau.

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A256035 Number of idempotent basis elements in partition monoid P_n.

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