A256034
Number of irreducible idempotents in partition monoid P_n.
Original entry on oeis.org
2, 8, 58, 648, 9794, 187302, 4353920, 119604518, 3803405406, 137828444548, 5621826966870, 255529007818470, 12836027705244956, 707657189518002658, 42563168959162893550, 2778631761757307345760, 196003207603955109742122
Offset: 1
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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 *)
A256035
Number of idempotent basis elements in partition monoid P_n.
Original entry on oeis.org
1, 1, 6, 59, 807, 14102, 301039, 7618613, 223586932, 7482796089, 281882090283
Offset: 0
A256042
Triangle read by rows: number of idempotent basis elements of rank k in partition monoid P_n.
Original entry on oeis.org
1, 0, 1, 0, 5, 1, 0, 43, 15, 1, 0, 529, 247, 30, 1, 0, 8451, 4795, 805, 50, 1, 0, 167397, 108871, 22710, 1985, 75, 1, 0, 3984807, 2855279, 697501, 76790, 4130, 105, 1, 0, 111319257, 85458479, 23520966, 3070501, 209930, 7658, 140, 1, 0, 3583777723, 2887069491, 871103269, 129732498, 10604811, 495054, 13062, 180, 1
Offset: 0
Triangle begins:
1,
0, 1,
0, 5, 1,
0, 43, 15, 1,
0, 529, 247, 30, 1,
0, 8451, 4795, 805, 50, 1,
0, 167397, 108871, 22710, 1985, 75, 1,
0, 3984807, 2855279, 697501, 76790, 4130, 105, 1,
0, 111319257, 85458479, 23520966, 3070501, 209930, 7658, 140, 1,
...
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rows = 10;
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];
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
M = BellMatrix[a33[# + 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, after Peter Luschny and R. J. Mathar *)
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# uses[bell_matrix from A264428]
A256042_generator = lambda n: A256033(n+1)
bell_matrix(A256042_generator, 9) # Peter Luschny, Jan 17 2016
Showing 1-3 of 3 results.
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