A178888 -id:A178888 - cp's OEIS frontend

A178886 Divide, cell by cell, sequence A178802 by sequence A178888.

1, 1, 1, 2, 2, 1, 6, 4, 2, 3, 1, 24, 12, 12, 6, 6, 4, 1, 120, 48, 48, 24, 18, 36, 6, 8, 12, 5, 1, 720, 240, 240, 240, 72, 144, 72, 72, 24, 72, 24, 10, 20, 6, 1, 5040, 1440, 1440, 1440, 720, 360

Offset: 1

Alford Arnold, Jun 21 2010

Sequences A178802, A178888 and A178886 have A000041(n) entries per row.

			A178802 begins 1 2 2 6 12 6 24 48 24 72 24 ...
A178888 begins 1 2 2 3 6 6 4 12 12 24 24 ...
therefore
A178886 begins 1 1 1 2 2 1 6 4 2 3 1 ...

Cf. A000041, A000522 (row sums), A002866, A178802, A178887, A178888.

T(n,k) = A178802(n,k) / A178888(n,k)

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

A178887 Total of n-colorings of parts of all integer partitions of n.

Original entry on oeis.org

1, 1, 4, 15, 76, 405, 2616, 18613, 151432, 1367649, 13720060, 151005261, 1812987804, 23570657773, 330012270784, 4950230221875, 79204352557936, 1346475340841553, 24236578276301844, 460495032000171373, 9209901462655990180, 193407932383031348241, 4254974546342806648384

Offset: 0

Alford Arnold, Jun 21 2010

An integer partition of n with k parts can have its parts colored in n!/(n-k)! ways. a(n) is the sum of all these possibilities over all integer partitions of n. - Olivier Gérard, May 08 2012

Table A178888 has A000041 entries per row.

			A178888 begins
1
2 2
3 6 6
4 12 12 24 24
...
therefore A178887 begins 1 4 15 76 405 ...

Alois P. Heinz, Table of n, a(n) for n = 0..449

Row sums of the irregular table A178888.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      p!/(p-n)!, b(n, i-1, p)+p*b(n-i, min(i, n-i), p-1))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2019

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, p!/(p - n)!, b[n, i - 1, p] + p b[n - i, Min[i, n - i], p - 1]];
a[n_] := b[n, n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

More terms and more direct definition by Olivier Gérard, May 08 2012

a(0)=1 prepended by Alois P. Heinz, Jan 21 2019

A179972 Irregular table T(n,k) = A178886(n,k)/A048996(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 2, 2, 1, 1, 24, 6, 6, 2, 2, 1, 1, 120, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1, 720, 120, 120, 120, 24, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1, 5040, 720, 720, 720, 720, 120, 120, 120, 120, 120, 24, 24, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1, 40320

Offset: 1

Alford Arnold, Aug 04 2010

Row n has A000041(n) terms.
Consider the five partitions of the number 4:
4 3+1 2+2 2+1+1 and 1+1+1+1
rewriting as 4000 3100 2200 2110 and 1111
then a(n) counts the ways that the zeros can be permuted:
6,2,2,1,1
agreeing with the factorial of the difference between A036042 and A036043.

			Row four of A178886 begins: 6 4 2 3 1
Row four of A048996 begins: 1 2 1 3 1
so,
Row four of A179972 begins: 6 2 2 1 1
Triangle T(n,k) begins:
    1;
    1,  1;
    2,  1,  1;
    6,  2,  2,  1, 1;
   24,  6,  6,  2, 2, 1, 1;
  120, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1;
  ...

Cf. A178886, A048996, A036042, A036043, A179973 (row sums).

T(n,k) = ( A036042(n,k) - A036043(n,k))!.

T(n,k) = n!/A178888(n,k). - R. J. Mathar, Mar 03 2011

cp's OEIS Frontend

A178886 Divide, cell by cell, sequence A178802 by sequence A178888.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A178887 Total of n-colorings of parts of all integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A179972 Irregular table T(n,k) = A178886(n,k)/A048996(n,k) read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula