A256034 Number of irreducible idempotents in partition monoid P_n.

2, 8, 58, 648, 9794, 187302, 4353920, 119604518, 3803405406, 137828444548, 5621826966870, 255529007818470, 12836027705244956, 707657189518002658, 42563168959162893550, 2778631761757307345760, 196003207603955109742122

Offset: 1

Author

N. J. A. Sloane, Mar 14 2015

Keywords

• nonn

Links

• I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014. See Table 3.

Crossrefs

Cf. A060639, A256033.

Programs

Mathematica

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]];
a33[n_] := Module[{b = 0}, Do[b += r s f[n, r, s], {r, n}, {s, n}]; b];
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]];
a[n_] := a33[n] + a39[n];
Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Dec 15 2018  *)

Formula

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