A244491 Number of minimal idempotent generating sets for the singular part P_n \ S_n of the partition monoid P_n.

1, 1, 3, 20, 201, 2604, 40915, 754368, 15960945, 381141008, 10139372451, 297356237760, 9530800099513, 331453265976000, 12430323314648499, 500046099516905984, 21478615942550889825, 981110493372418629888, 47489191763845877910595

Offset: 0

Author

N. J. A. Sloane, Jul 05 2014

Keywords

• nonn

Links

• J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014-2016.

Programs

Maple

A038205 := proc(n)
    option remember ;
    if n = 0 then
        1;
    elif n <=2 then
        0 ;
    else
        (n-1)*procname(n-1)+(n-1)*(n-2)*procname(n-3) ;
    end if;
end proc:
A244490 := proc(n,k)
    add((-1)^i*binomial(k,2*i)*doublefactorial(2*i-1)*n^(k-2*i),i=0..floor(k/2)) ;
end proc:
A244491 := proc(n)
    add(binomial(n,k)*A038205(k)*A244490(n,n-k),k=0..n) ;
end proc:
seq(A244491(n),n=0..30) ; # R. J. Mathar, Aug 26 2014

Mathematica

a05[n_] := SeriesCoefficient[Exp[-x - x^2/2]/(1 - x), {x, 0, n}]*n!;
a90[n_, k_] := Sum[(-1)^i*Binomial[k, 2i]*(2i-1)!!*n^(k-2*i), {i, 0, k/2}];
a[n_] := Sum[Binomial[n, k]*a05[k]*a90[n, n - k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 01 2017, after R. J. Mathar *)

Formula

An explicit formula is given in Th. 7.13 of East-Gray.