A049009 - OEIS

%I #31 Jan 18 2022 06:00:16

%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 n-th 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

%C Let f,g be functions from [n] into [n]. Let S_n be the symmetric group on n letters. Then f and g form the same partition iff S_nfS_n = S_ngS_n. - _Geoffrey Critzer_, Jan 13 2022

%D O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page38.

%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.

%t f[list_] := Multinomial @@ Join[{nn - Length[list]}, Table[Count[list, i], {i, 1, nn}]]*Multinomial @@ list; Table[nn = n; Map[f, IntegerPartitions[nn]], {n, 0, 10}] // Grid (* _Geoffrey Critzer_, Jan 13 2022 *)

%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. Adams-Watters_, May 30 2006

%E a(0)=1 prepended by _Andrew Howroyd_, Oct 18 2020