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A048793 List giving all subsets of natural numbers arranged in standard statistical (or Yates) order.

%I A048793 #33 Feb 01 2023 14:34:08
%S A048793 0,1,2,1,2,3,1,3,2,3,1,2,3,4,1,4,2,4,1,2,4,3,4,1,3,4,2,3,4,1,2,3,4,5,
%T A048793 1,5,2,5,1,2,5,3,5,1,3,5,2,3,5,1,2,3,5,4,5,1,4,5,2,4,5,1,2,4,5,3,4,5,
%U A048793 1,3,4,5,2,3,4,5,1,2,3,4,5,6,1,6,2,6,1,2,6,3,6,1,3,6,2,3,6,1,2,3,6,4,6,1,4
%N A048793 List giving all subsets of natural numbers arranged in standard statistical (or Yates) order.
%C A048793 For n>0: first occurrence of n in row 2^(n-1), and when the table is seen as a flattened list at position n*2^(n-1)+1, cf. A005183. - _Reinhard Zumkeller_, Nov 16 2013
%C A048793 Row n lists the positions of 1's in the reversed binary expansion of n. Compare to triangles A112798 and A213925. - _Gus Wiseman_, Jul 22 2019
%D A048793 S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, p. 249.
%H A048793 Reinhard Zumkeller, <a href="/A048793/b048793.txt">Rows n = 0..1000 of triangle, flattened</a>
%F A048793 Constructed recursively: subsets that include n are obtained by appending n to all earlier subsets.
%e A048793 From _Gus Wiseman_, Jul 22 2019: (Start)
%e A048793 Triangle begins:
%e A048793   {}
%e A048793   1
%e A048793   2
%e A048793   1  2
%e A048793   3
%e A048793   1  3
%e A048793   2  3
%e A048793   1  2  3
%e A048793   4
%e A048793   1  4
%e A048793   2  4
%e A048793   1  2  4
%e A048793   3  4
%e A048793   1  3  4
%e A048793   2  3  4
%e A048793   1  2  3  4
%e A048793   5
%e A048793   1  5
%e A048793   2  5
%e A048793   1  2  5
%e A048793   3  5
%e A048793 (End)
%p A048793 T:= proc(n) local i, l, m; l:= NULL; m:= n;
%p A048793       if n=0 then return 0 fi; for i while m>0 do
%p A048793       if irem(m, 2, 'm')=1 then l:=l, i fi od; l
%p A048793     end:
%p A048793 seq(T(n), n=0..50);  # _Alois P. Heinz_, Sep 06 2014
%t A048793 s[0] = {{}}; s[n_] := s[n] = Join[s[n - 1], Append[#, n]& /@ s[n - 1]]; Join[{0}, Flatten[s[6]]] (* _Jean-François Alcover_, May 24 2012 *)
%t A048793 Table[Join@@Position[Reverse[IntegerDigits[n,2]],1],{n,30}] (* _Gus Wiseman_, Jul 22 2019 *)
%o A048793 (C)
%o A048793 #include <stdio.h>
%o A048793 #include <stdlib.h>
%o A048793 #define USAGE "Usage: 'A048793 num' where num is the largest number to use creating sets.\n"
%o A048793 #define MAX_NUM 10
%o A048793 #define MAX_ROW 1024
%o A048793 int main(int argc, char *argv[]) {
%o A048793   unsigned short a[MAX_ROW][MAX_NUM]; signed short old_row, new_row, i, j, end;
%o A048793   if (argc < 2) { fprintf(stderr, USAGE); return EXIT_FAILURE; }
%o A048793   end = atoi(argv[1]); end = (end > MAX_NUM) ? MAX_NUM: end;
%o A048793   for (i = 0; i < MAX_ROW; i++) for ( j = 0; j < MAX_NUM; j++) a[i][j] = 0;
%o A048793   a[1][0] = 1; new_row = 2;
%o A048793   for (i = 2; i <= end; i++) {
%o A048793     a[new_row++ ][0] = i;
%o A048793     for (old_row = 1; a[old_row][0] != i; old_row++) {
%o A048793       for (j = 0; a[old_row][j] != 0; j++) { a[new_row][j] = a[old_row][j]; }
%o A048793       a[new_row++ ][j] = i;
%o A048793     }
%o A048793   }
%o A048793   fprintf(stdout, "Values: 0");
%o A048793   for (i = 1; a[i][0] != 0; i++) for (j = 0; a[i][j] != 0; j++) fprintf(stdout, ",%d", a[i][j]);
%o A048793   fprintf(stdout, "\n"); return EXIT_SUCCESS
%o A048793 }
%o A048793 (Haskell)
%o A048793 a048793 n k = a048793_tabf !! n !! k
%o A048793 a048793_row n = a048793_tabf !! n
%o A048793 a048793_tabf = [0] : [1] : f [[1]] where
%o A048793    f xss = yss ++ f (xss ++ yss) where
%o A048793      yss = [y] : map (++ [y]) xss
%o A048793      y = last (last xss) + 1
%o A048793 -- _Reinhard Zumkeller_, Nov 16 2013
%Y A048793 Cf. A048794.
%Y A048793 Row lengths are A000120.
%Y A048793 First column is A001511.
%Y A048793 Heinz numbers of rows are A019565.
%Y A048793 Row sums are A029931.
%Y A048793 Reversing rows gives A272020.
%Y A048793 Subtracting 1 from each term gives A133457; subtracting 1 and reversing rows gives A272011.
%Y A048793 Indices of relatively prime rows are A291166 (see also A326674); arithmetic progressions are A295235; rows with integer average are A326669 (see also A326699/A326700); pairwise coprime rows are A326675.
%Y A048793 Cf. A035327, A070939.
%K A048793 nonn,tabf,easy,nice
%O A048793 0,3
%A A048793 _N. J. A. Sloane_
%E A048793 More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2000