A035796 -id:A035796 - cp's OEIS frontend

A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

0, 1, 2, 2, 3, 3, 4, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 7, 6, 5, 7, 4, 6, 6, 8, 7, 6, 8, 5, 7, 7, 9, 8, 7, 9, 6, 8, 6, 8, 10, 7, 9, 6, 8, 8, 10, 7, 9, 7, 9, 11, 8, 10, 5, 7, 9, 9, 11, 8, 10, 8, 10, 12, 9, 11, 6, 8, 10, 8, 10, 12, 7, 9, 9, 11, 9, 8, 11, 10, 13, 10, 12, 7, 9, 11, 9, 11, 13, 8, 10, 10, 12

Offset: 1

Alford Arnold

			a(3) = 2 since A025487(3) = 4 = 2*2; a(5) = 3 since A025487(5) = 8 = 2*2*2; ...

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A025487, A000041, A035796, A036042, A001222.

a(n) = A001222(A025487(n)) = A001222(A181822(n)).

More terms from Henry Bottomley, Apr 30 2001
Edited to accommodate change in A025487's offset by Matthew Vandermast, Nov 08 2008
Definition corrected by Álvar Ibeas, Nov 01 2014

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.

Original entry on oeis.org

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

Offset: 0

Alford Arnold

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

			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.

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

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].

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

Row sizes A000041, sums A000312.

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 *)

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

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

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

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

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A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

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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.

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