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
KEYWORD
nonn
AUTHOR
Christian G. Bower, Apr 01 1998
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Sep 02 2015
STATUS
approved