|
%I #18 Jan 04 2022 14:15:26
%S 1,1,1,1,6,2,1,21,36,6,1,60,300,240,24,1,155,1800,3900,1800,120,1,378,
%T 9030,42000,50400,15120,720,1,889,40572,357210,882000,670320,141120,
%U 5040,1,2040,169400,2610720,11677680,17781120,9313920,1451520,40320
%N Triangle read by rows: (1/n)*T(n,h), where T(n,h) is the array in A101817.
%C Column 2 is A066524.
%C T(n,h) is the number of partial functions f:{1,2,...,n-1}->{1,2,...,n-1} such that |Image(f)| = h-1. Equivalently T(n,h) = |D_h(a)| where D_h(a) is Green's D-class containing a, with a in the semigroup of partial transformations on [n-1] and rank(a) = h-1. - _Geoffrey Critzer_, Jan 02 2022
%D O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, page 61.
%F T(n, h) = (1/n)*C(n, h)*U(n, h), where U(n, h) is the array in A019538.
%F T(n, h) = Stirling2(n,h)*(n-1)!/(n-h)!. - _Geoffrey Critzer_, Jan 02 2022
%e First rows:
%e 1
%e 1 1
%e 1 6 2
%e 1 21 36 6
%t Table[Table[StirlingS2[n, k] (n-1)!/(n - k)!, {k, 1, n}], {n, 1,
%t 6}] // Grid (* _Geoffrey Critzer_, Jan 02 2022 *)
%Y Cf. A019538, A066524, A090657, A101817, A101819, A101820, A101821, A000169 (row sums).
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Dec 17 2004
%E Offset changed to 1 by _Alois P. Heinz_, Jan 03 2022
|
