A120774 Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.

1, 1, 2, 8, 31, 147, 899, 5777, 41024, 322488, 2749325, 25118777, 245389896, 2554780438, 28009868787, 323746545433, 3933023224691, 49924332801387, 661988844566017, 9138403573970063, 131043199040556235, 1949750421507432009, 30031656711776544610

Offset: 0

Author

Alford Arnold, Jul 12 2006

Keywords

• easy
• nonn

Comments

Old name was: Row sums of A179233.

a(n) is the number of ways to linearly order the blocks in each set partition of {1,2,...,n} where two blocks are considered identical if they have the same number of elements. - Geoffrey Critzer, Sep 29 2011

Examples

			A179233 begins 1; 1; 1 1; 6 1 1; 8 3 18 1 1 ... with row sums 1, 1 2 8 31 147 ...
a(3) = 8: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3, 1|2|3. - _Alois P. Heinz_, Apr 27 2017
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..525

Crossrefs

Cf. A000041, A032011, A049019, A096161, A096162, A196301.

Row sums of A179233, A285824.

Main diagonal of A327244.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      (p+n)!/n!, add(b(n-i*j, i-1, p+j)*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 27 2017

Mathematica

f[{x_,y_}]:= x!^y y!;   Table[Total[Table[n!,{PartitionsP[n]}]/Apply[Times,Map[f,Map[Tally,Partitions[n]],{2}],2] * Apply[Multinomial,Map[Last,Map[Tally,Partitions[n]],{2}],2]],{n,0,20}]  (* Geoffrey Critzer, Sep 29 2011 *)

Extensions

Leading 1 inserted, definition simplified by R. J. Mathar, Sep 28 2011

a(15) corrected, more terms, and new name (using Geoffrey Critzer's comment) from Alois P. Heinz, Apr 27 2017