cp's OEIS Frontend

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

%I A244491 #14 Dec 01 2017 09:38:32
%S A244491 1,1,3,20,201,2604,40915,754368,15960945,381141008,10139372451,
%T A244491 297356237760,9530800099513,331453265976000,12430323314648499,
%U A244491 500046099516905984,21478615942550889825,981110493372418629888,47489191763845877910595
%N A244491 Number of minimal idempotent generating sets for the singular part P_n \ S_n of the partition monoid P_n.
%H A244491 J. East, R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014-2016.
%F A244491 An explicit formula is given in Th. 7.13 of East-Gray.
%p A244491 A038205 := proc(n)
%p A244491     option remember ;
%p A244491     if n = 0 then
%p A244491         1;
%p A244491     elif n <=2 then
%p A244491         0 ;
%p A244491     else
%p A244491         (n-1)*procname(n-1)+(n-1)*(n-2)*procname(n-3) ;
%p A244491     end if;
%p A244491 end proc:
%p A244491 A244490 := proc(n,k)
%p A244491     add((-1)^i*binomial(k,2*i)*doublefactorial(2*i-1)*n^(k-2*i),i=0..floor(k/2)) ;
%p A244491 end proc:
%p A244491 A244491 := proc(n)
%p A244491     add(binomial(n,k)*A038205(k)*A244490(n,n-k),k=0..n) ;
%p A244491 end proc:
%p A244491 seq(A244491(n),n=0..30) ; # _R. J. Mathar_, Aug 26 2014
%t A244491 a05[n_] := SeriesCoefficient[Exp[-x - x^2/2]/(1 - x), {x, 0, n}]*n!;
%t A244491 a90[n_, k_] := Sum[(-1)^i*Binomial[k, 2i]*(2i-1)!!*n^(k-2*i), {i, 0, k/2}];
%t A244491 a[n_] := Sum[Binomial[n, k]*a05[k]*a90[n, n - k], {k, 0, n}];
%t A244491 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Dec 01 2017, after _R. J. Mathar_ *)
%K A244491 nonn
%O A244491 0,3
%A A244491 _N. J. A. Sloane_, Jul 05 2014