A256042 Triangle read by rows: number of idempotent basis elements of rank k in partition monoid P_n.

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

Author

N. J. A. Sloane, Mar 14 2015

Keywords

• nonn
• tabl

Comments

Also the Bell transform of A256033(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 17 2016

Examples

			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,
...
		

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.

Crossrefs

Cf. A256033.

Programs

Mathematica

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 *)

Sage

# uses[bell_matrix from A264428]
A256042_generator = lambda n: A256033(n+1)
bell_matrix(A256042_generator, 9) # Peter Luschny, Jan 17 2016