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A032011
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Partition n labeled elements into sets of different sizes and order the sets.
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15
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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
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OFFSET
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0,4
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COMMENTS
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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)
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LINKS
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FORMULA
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"AGJ" (ordered, elements, labeled) transform of 1, 1, 1, 1, ...
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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