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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. 9
 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 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 REFERENCES O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page38. 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. MATHEMATICA 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 *) 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 September 15 02:42 EDT 2024. Contains 375930 sequences. (Running on oeis4.)