%I
%S 1,1,2,2,3,18,6,4,48,36,144,24,5,100,200,600,900,1200,120,6,180,450,
%T 300,1800,7200,1800,7200,16200,10800,720,7,294,882,1470,4410,22050,
%U 14700,22050,29400,176400,88200,88200,264600,105840,5040,8,448,1568,3136,1960
%N Number of functions from a set to itself such that the sizes of the preimages of the individual elements in the range form the nth partition in Abramowitz and Stegun order.
%C a(n,k) is a refinement of 1; 2,2; 3,18,6; 4,84,144,24; ... cf. A019575.
%C a(n,k)/A036040(n,k) and a(n,k)/A048996(n,k) are also integer sequences.
%C Apparently a(n,k)/A036040(n,k) = A178888(n,k).  R. J. Mathar, Apr 17 2011
%H Andrew Howroyd, <a href="/A049009/b049009.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F a(n,k) = A036038(n,k) * A035206(n,k).
%e Table begins:
%e 1;
%e 1;
%e 2, 2;
%e 3, 18, 6;
%e 4, 48, 36, 144, 24;
%e ...
%e For n = 4, partition [3], we can map all three of {1,2,3} to any one of them, for 3 possible values. For n=5, partition [2,1], there are 3 choices for which element is alone in a preimage, 3 choices for which element to map that to and then 2 choices for which element to map the pair to, so a(5) = 3*3*2 = 18.
%o (PARI)
%o C(sig)={my(S=Set(sig)); (binomial(vecsum(sig), #sig)) * (#sig)! * vecsum(sig)! / (prod(k=1, #S, (#select(t>t==S[k], sig))!) * prod(k=1, #sig, sig[k]!))}
%o Row(n)={apply(C, [Vecrev(p)  p<partitions(n)])}
%o { for(n=0, 7, print(Row(n))) } \\ _Andrew Howroyd_, Oct 18 2020
%Y Cf. A019575, A035206, A035796, A036038, A036040, A048996.
%Y Row sizes A000041, sums A000312.
%K nonn,tabf,easy
%O 0,3
%A _Alford Arnold_
%E Better definition from _Franklin T. AdamsWatters_, May 30 2006
%E a(0)=1 prepended by _Andrew Howroyd_, Oct 18 2020
