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

%I A032011 #44 Sep 13 2018 18:32:23
%S A032011 1,1,1,7,9,31,403,757,2873,12607,333051,761377,3699435,16383121,
%T A032011 108710085,4855474267,13594184793,76375572751,388660153867,
%U A032011 2504206435681,20148774553859,1556349601444477,5050276538344665,33326552998257031,186169293932977115,1305062351972825281,9600936552132048553,106019265737746665727,12708226588208611056333,47376365554715905155127
%N A032011 Partition n labeled elements into sets of different sizes and order the sets.
%C A032011 From _Alois P. Heinz_, Sep 02 2015: (Start)
%C A032011 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.
%C A032011 a(3) = 7:
%C A032011 [1]   [1 0]  [0 1]  [1 0]  [0 1]  [0 1]  [1 0]
%C A032011 [1]   [1 0]  [0 1]  [0 1]  [1 0]  [1 0]  [0 1]
%C A032011 [1]   [0 1]  [1 0]  [1 0]  [0 1]  [1 0]  [0 1].
%C A032011 3abc, 2ab1c, 1c2ab, 2ac1b, 1b2ac, 2bc1a, 1a2bc.  (End)
%H A032011 Alois P. Heinz, <a href="/A032011/b032011.txt">Table of n, a(n) for n = 0..670</a>
%H A032011 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A032011 "AGJ" (ordered, elements, labeled) transform of 1, 1, 1, 1, ...
%F A032011 a(n) = Sum_{k>=0} k! * A131632(n,k). - _Alois P. Heinz_, Sep 09 2015
%p A032011 b:= proc(n, i, p) option remember;
%p A032011       `if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p)+
%p A032011       `if`(i>n, 0, b(n-i, i-1, p+1)*binomial(n,i))))
%p A032011     end:
%p A032011 a:= n-> b(n$2, 0):
%p A032011 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 02 2015
%t A032011 f[list_]:=Apply[Multinomial,list]*Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 1, 30}]
%t A032011 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_ *)
%o A032011 (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
%Y A032011 Cf. A000670, A007837, A032020, A114902, A120774, A131632.
%Y A032011 Main diagonal of A261836 and A261959.
%K A032011 nonn
%O A032011 0,4
%A A032011 _Christian G. Bower_, Apr 01 1998
%E A032011 a(0)=1 prepended by _Alois P. Heinz_, Sep 02 2015