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A256041 Triangle read by rows: number of idempotent basis elements of rank k in Brauer monoid B_n.

%I A256041 #18 Aug 06 2019 07:35:04
%S A256041 1,0,1,0,0,1,0,6,0,1,0,0,24,0,1,0,120,0,60,0,1,0,0,1080,0,120,0,1,0,
%T A256041 5040,0,5040,0,210,0,1,0,0,80640,0,16800,0,336,0,1,0,362880,0,604800,
%U A256041 0,45360,0,504,0,1,0,0,9072000,0,3024000,0,105840,0,720,0,1
%N A256041 Triangle read by rows: number of idempotent basis elements of rank k in Brauer monoid B_n.
%C A256041 Also the Bell transform of A005212(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%H A256041 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 A256041 Triangle begins:
%e A256041 1,
%e A256041 0, 1,
%e A256041 0, 0, 1,
%e A256041 0, 6, 0, 1,
%e A256041 0, 0, 24, 0, 1,
%e A256041 0, 120, 0, 60, 0, 1,
%e A256041 0, 0, 1080, 0, 120, 0, 1,
%e A256041 0, 5040, 0, 5040, 0, 210, 0, 1,
%e A256041 0, 0, 80640, 0, 16800, 0, 336, 0, 1,
%e A256041 0, 362880, 0, 604800, 0, 45360, 0, 504, 0, 1,
%e A256041 0, 0, 9072000, 0, 3024000, 0, 105840, 0, 720, 0, 1,
%e A256041 ...
%p A256041 # The function BellMatrix is defined in A264428.
%p A256041 BellMatrix(n -> `if`(n::odd, 0, (n+1)!), 9); # _Peter Luschny_, Jan 29 2016
%t A256041 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A256041 B = BellMatrix[Function[n, If[OddQ[n], 0, (n + 1)!]], rows = 12];
%t A256041 Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y A256041 Cf. A005212, A264428.
%K A256041 nonn,tabl
%O A256041 0,8
%A A256041 _N. J. A. Sloane_, Mar 14 2015