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 A032011 Partition n labeled elements into sets of different sizes and order the sets. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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)/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 *) 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

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Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)