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Number of ordered set partitions of [n] where equal-sized blocks are ordered with increasing least elements.
6

%I #28 Sep 14 2019 16:13:05

%S 1,1,2,8,31,147,899,5777,41024,322488,2749325,25118777,245389896,

%T 2554780438,28009868787,323746545433,3933023224691,49924332801387,

%U 661988844566017,9138403573970063,131043199040556235,1949750421507432009,30031656711776544610

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

%C Old name was: Row sums of A179233.

%C 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

%H Alois P. Heinz, <a href="/A120774/b120774.txt">Table of n, a(n) for n = 0..525</a>

%e A179233 begins 1; 1; 1 1; 6 1 1; 8 3 18 1 1 ... with row sums 1, 1 2 8 31 147 ...

%e 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

%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1,

%p (p+n)!/n!, add(b(n-i*j, i-1, p+j)*combinat

%p [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Apr 27 2017

%t 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 *)

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

%Y Row sums of A179233, A285824.

%Y Main diagonal of A327244.

%K easy,nonn

%O 0,3

%A _Alford Arnold_, Jul 12 2006

%E Leading 1 inserted, definition simplified by _R. J. Mathar_, Sep 28 2011

%E a(15) corrected, more terms, and new name (using _Geoffrey Critzer_'s comment) from _Alois P. Heinz_, Apr 27 2017