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%I #22 Mar 24 2020 05:06:06
%S 1,0,1,0,5,1,0,43,15,1,0,529,247,30,1,0,8451,4795,805,50,1,0,167397,
%T 108871,22710,1985,75,1,0,3984807,2855279,697501,76790,4130,105,1,0,
%U 111319257,85458479,23520966,3070501,209930,7658,140,1,0,3583777723,2887069491,871103269,129732498,10604811,495054,13062,180,1
%N Triangle read by rows: number of idempotent basis elements of rank k in partition monoid P_n.
%C Also the Bell transform of A256033(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 17 2016
%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.
%e Triangle begins:
%e 1,
%e 0, 1,
%e 0, 5, 1,
%e 0, 43, 15, 1,
%e 0, 529, 247, 30, 1,
%e 0, 8451, 4795, 805, 50, 1,
%e 0, 167397, 108871, 22710, 1985, 75, 1,
%e 0, 3984807, 2855279, 697501, 76790, 4130, 105, 1,
%e 0, 111319257, 85458479, 23520966, 3070501, 209930, 7658, 140, 1,
%e ...
%t rows = 10;
%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 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t M = BellMatrix[a33[# + 1]&, rows];
%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 19 2018, after _Peter Luschny_ and _R. J. Mathar_ *)
%o (Sage) # uses[bell_matrix from A264428]
%o A256042_generator = lambda n: A256033(n+1)
%o bell_matrix(A256042_generator, 9) # _Peter Luschny_, Jan 17 2016
%Y Cf. A256033.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Mar 14 2015
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