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A049009 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. 8
1, 1, 2, 2, 3, 18, 6, 4, 48, 36, 144, 24, 5, 100, 200, 600, 900, 1200, 120, 6, 180, 450, 300, 1800, 7200, 1800, 7200, 16200, 10800, 720, 7, 294, 882, 1470, 4410, 22050, 14700, 22050, 29400, 176400, 88200, 88200, 264600, 105840, 5040, 8, 448, 1568, 3136, 1960 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n,k) is a refinement of 1; 2,2; 3,18,6; 4,84,144,24; ... cf. A019575.

a(n,k)/A036040(n,k) and a(n,k)/A048996(n,k) are also integer sequences.

Apparently a(n,k)/A036040(n,k) = A178888(n,k). - R. J. Mathar, Apr 17 2011

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

a(n,k) = A036038(n,k) * A035206(n,k).

EXAMPLE

Table begins:

  1;

  1;

  2,  2;

  3, 18,  6;

  4, 48, 36, 144, 24;

  ...

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.

PROG

(PARI)

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]!))}

Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}

{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

CROSSREFS

Cf. A019575, A035206, A035796, A036038, A036040, A048996.

Row sizes A000041, sums A000312.

Sequence in context: A089751 A137909 A035796 * A101817 A058159 A058157

Adjacent sequences:  A049006 A049007 A049008 * A049010 A049011 A049012

KEYWORD

nonn,tabf,easy

AUTHOR

Alford Arnold

EXTENSIONS

Better definition from Franklin T. Adams-Watters, May 30 2006

a(0)=1 prepended by Andrew Howroyd, Oct 18 2020

STATUS

approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)