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A256034 Number of irreducible idempotents in partition monoid P_n. 2
2, 8, 58, 648, 9794, 187302, 4353920, 119604518, 3803405406, 137828444548, 5621826966870, 255529007818470, 12836027705244956, 707657189518002658, 42563168959162893550, 2778631761757307345760, 196003207603955109742122 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..17.

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.

FORMULA

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

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-Fran├žois Alcover, Dec 15 2018  *)

CROSSREFS

Cf. A060639, A256033.

Sequence in context: A005804 A162067 A179534 * A086907 A132186 A191603

Adjacent sequences:  A256031 A256032 A256033 * A256035 A256036 A256037

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mar 14 2015

STATUS

approved

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Last modified March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)