A090683 - OEIS

%I #31 Dec 28 2022 02:23:06

%S 1,0,1,0,2,1,0,3,9,1,0,4,42,24,1,0,5,150,250,50,1,0,6,465,1800,975,90,

%T 1,0,7,1323,10535,12250,2940,147,1,0,8,3556,54096,119070,58800,7448,

%U 224,1,0,9,9180,254100,979020,875826,222264,16632,324,1,0,10,22995,1119600,7162050,10716300,4793670,705600,33750,450,1

%N Triangle read by rows, defined by T(n,k) = C(n,k)*S2(n,k), 0 <= k <= n, where C(n,k) are the binomial coefficients and S2(n,k) are the Stirling numbers of the second kind.

%C T(n,k) is the number of Green's H-classes contained in the D-class of rank k in the full transformation semigroup on [n]. - _Geoffrey Critzer_, Dec 27 2022

%D O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, pages 58-62.

%H Vincenzo Librandi, <a href="/A090683/b090683.txt">Table of n, a(n) for n = 0..230</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Green&#39;s_relations">Green's relations</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Transformation_semigroup">Transformation semigroup</a>

%F T(n, k) = binomial(n,k)*Stirling2(n,k).

%F T(n, k) = A007318(n, k)*A048993(n, k).

%F T(n, k) = A090657(n, k)/k!.

%e Triangle begins:

%e   1;

%e   0,  1;

%e   0,  2,  1;

%e   0,  3,  9,  1;

%e   0,  4, 42, 24,  1;

%e   ...

%t Flatten[Table[Table[Binomial[n, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 10}], 1]

%o (Maxima) create_list(binomial(n,k)*stirling2(n,k),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */

%Y Cf. A007318, A048993, A090657.

%Y Row sum sequence is A122455.

%K easy,nonn,tabl

%O 0,5

%A _Philippe Deléham_, Dec 18 2003

%E Edited by _Olivier Gérard_, Oct 23 2012