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

0,4

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)

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

Maple

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p)+
      `if`(i>n, 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)/2<n, 0, If[n==0, p!, b[n, i-1, p] + If[i>n, 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 *)

Prog

(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

Crossrefs

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

Main diagonal of A261836 and A261959.

Sequence in context: A147248 A147186 A131623 * A272434 A272433 A272432

Adjacent sequences: A032008 A032009 A032010 * A032012 A032013 A032014

Keyword

nonn

Author

Christian G. Bower, Apr 01 1998

Extensions

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

Status

approved