A256034 - OEIS

%I #13 Dec 15 2018 11:50:25

%S 2,8,58,648,9794,187302,4353920,119604518,3803405406,137828444548,

%T 5621826966870,255529007818470,12836027705244956,707657189518002658,

%U 42563168959162893550,2778631761757307345760,196003207603955109742122

%N Number of irreducible idempotents in partition monoid P_n.

%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., <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 Table 3.

%F a(n) = A060639(n) + A256033(n).

%t f[n_, r_, s_] := f[n, r, s] = Module[{resu, m, a, b}, Which[n <= 0, 0, s == 1, StirlingS2[n, r], r == 1, StirlingS2[n, s], True, resu = s f[n-1, r-1, s] + r f[n-1, r, s-1] + r s f[n-1, r, s]; Do[resu += Binomial[n-2, m] (b (r-a) + a (s-b)) f[m, a, b] f[-m+n-1, r-a, s-b], {m, n}, {a, r-1}, {b, s-1}]; resu]];

%t a33[n_] := Module[{b = 0}, Do[b += r s f[n, r, s], {r, n}, {s, n}]; b];

%t a39[n_] := Module[{t}, t = Table[BellB[k-1]^2/(k-1)!, {k, 1, n+1}]; n! SeriesCoefficient[1 + Log[O[x]^(n+1) + Sum[t[[i]] x^(i-1), {i, 1, Length[t]}]], n]];

%t a[n_] := a33[n] + a39[n];

%t Table[a[n], {n, 1, 17}] (* _Jean-François Alcover_, Dec 15 2018  *)

%Y Cf. A060639, A256033.

%K nonn

%O 1,1

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