A032011 Partition n labeled elements into sets of different sizes and order the sets.

1, 1, 1, 7, 9, 31, 403, 757, 2873, 12607, 333051, 761377, 3699435, 16383121, 108710085, 4855474267, 13594184793, 76375572751, 388660153867, 2504206435681, 20148774553859, 1556349601444477, 5050276538344665, 33326552998257031, 186169293932977115, 1305062351972825281, 9600936552132048553, 106019265737746665727, 12708226588208611056333, 47376365554715905155127

Offset: 0

Author

Christian G. Bower, Apr 01 1998

Keywords

• nonn

Comments

From Alois P. Heinz, Sep 02 2015: (Start)

Also the number of matrices with n rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct. Equivalently, the number of compositions of n into distinct parts where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once.

a(3) = 7:

[1] [1 0] [0 1] [1 0] [0 1] [0 1] [1 0]

[1] [1 0] [0 1] [0 1] [1 0] [1 0] [0 1]

[1] [0 1] [1 0] [1 0] [0 1] [1 0] [0 1].

3abc, 2ab1c, 1c2ab, 2ac1b, 1b2ac, 2bc1a, 1a2bc. (End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..670
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A000670, A007837, A032020, A114902, A120774, A131632.

Main diagonal of A261836 and A261959.

Programs

Maple

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1)*binomial(n,i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 02 2015

Mathematica

f[list_]:=Apply[Multinomial,list]*Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 1, 30}]
b[n_, i_, p_] := b[n, i, p] = If[i*(i+1)/2n, 0, b[n-i, i-1, p+1]*Binomial[n, i]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

PARI

seq(n)=[subst(serlaplace(y^0*p),y,1) | p <- Vec(serlaplace(prod(k=1, n, 1 + x^k*y/k! + O(x*x^n))))] \\ Andrew Howroyd, Sep 13 2018

Formula

"AGJ" (ordered, elements, labeled) transform of 1, 1, 1, 1, ...

a(n) = Sum_{k>=0} k! * A131632(n,k). - Alois P. Heinz, Sep 09 2015

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 02 2015