A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

2, 3, 12, 30, 240, 840, 10080, 45360, 725760, 3991680, 79833600, 518918400, 12454041600, 93405312000, 2615348736000, 22230464256000, 711374856192000, 6758061133824000, 243290200817664000, 2554547108585472000, 102181884343418880000, 1175091669949317120000

Offset: 1

Author

N. J. A. Sloane, Mar 14 2015

Keywords

• nonn

Comments

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

Links

• I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde and N. Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014. See Prop. 22.

Crossrefs

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

Programs

Maple

A256031 := proc(n)
    if type(n,'odd') then
        2*n! ;
    else
        (n+1)*(n-1)! ;
    end if;
end proc:
seq(A256031(n),n=1..20) ; # R. J. Mathar, Mar 14 2015

Mathematica

a[n_] := If[OddQ[n], 2*n!, (n + 1)*(n - 1)!];
Array[a, 20] (* Jean-François Alcover, Nov 24 2017, from Maple *)

Formula

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

a(2n-1) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A208529(2n+1). - Omar E. Pol, Mar 14 2015

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