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Number of irreducible idempotents in partial Brauer monoid PB_n.
3

%I #26 Feb 02 2023 02:24:16

%S 2,3,12,30,240,840,10080,45360,725760,3991680,79833600,518918400,

%T 12454041600,93405312000,2615348736000,22230464256000,711374856192000,

%U 6758061133824000,243290200817664000,2554547108585472000,102181884343418880000,1175091669949317120000

%N Number of irreducible idempotents in partial Brauer monoid PB_n.

%C Table 2 in chapter 7 of the preprint contains a typo: a(9) is not 725860. - _R. J. Mathar_, Mar 14 2015

%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde and N. Loughlin, <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv preprint arXiv:1408.2021 [math.GR], 2014. See Prop. 22.

%F There are simple formulas for the two bisections - see Dolinka et al.

%F a(2n-1) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A208529(2n+1). - _Omar E. Pol_, Mar 14 2015

%F Sum_{n>=1} 1/a(n) = (e^2+3)/(4*e) = 1/e + sinh(1)/2. - _Amiram Eldar_, Feb 02 2023

%p A256031 := proc(n)

%p if type(n,'odd') then

%p 2*n! ;

%p else

%p (n+1)*(n-1)! ;

%p end if;

%p end proc:

%p seq(A256031(n),n=1..20) ; # _R. J. Mathar_, Mar 14 2015

%t a[n_] := If[OddQ[n], 2*n!, (n + 1)*(n - 1)!];

%t Array[a, 20] (* _Jean-François Alcover_, Nov 24 2017, from Maple *)

%Y Cf. A052612, A052616, A052849, A098558, A208529, A256032, A174549 (bisection).

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Mar 14 2015