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

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• easy
• nice

Comments

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

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

Examples

			From _Gus Wiseman_, Jul 22 2019: (Start)
Triangle begins:
  {}
  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
  1  5
  2  5
  1  2  5
  3  5
(End)
		

References

• S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer-Verlag, NY, 1999, p. 249.

Links

• Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Crossrefs

Cf. A048794.

Row lengths are A000120.

First column is A001511.

Heinz numbers of rows are A019565.

Row sums are A029931.

Reversing rows gives A272020.

Subtracting 1 from each term gives A133457; subtracting 1 and reversing rows gives A272011.

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.

Cf. A035327, A070939.

Programs

C

#include 
#include 
#define USAGE "Usage: 'A048793 num' where num is the largest number to use creating sets.\n"
#define MAX_NUM 10
#define MAX_ROW 1024
int main(int argc, char *argv[]) {
  unsigned short a[MAX_ROW][MAX_NUM]; signed short old_row, new_row, i, j, end;
  if (argc < 2) { fprintf(stderr, USAGE); return EXIT_FAILURE; }
  end = atoi(argv[1]); end = (end > MAX_NUM) ? MAX_NUM: end;
  for (i = 0; i < MAX_ROW; i++) for ( j = 0; j < MAX_NUM; j++) a[i][j] = 0;
  a[1][0] = 1; new_row = 2;
  for (i = 2; i <= end; i++) {
    a[new_row++ ][0] = i;
    for (old_row = 1; a[old_row][0] != i; old_row++) {
      for (j = 0; a[old_row][j] != 0; j++) { a[new_row][j] = a[old_row][j]; }
      a[new_row++ ][j] = i;
    }
  }
  fprintf(stdout, "Values: 0");
  for (i = 1; a[i][0] != 0; i++) for (j = 0; a[i][j] != 0; j++) fprintf(stdout, ",%d", a[i][j]);
  fprintf(stdout, "\n"); return EXIT_SUCCESS
}

Haskell

a048793 n k = a048793_tabf !! n !! k
a048793_row n = a048793_tabf !! n
a048793_tabf = [0] : [1] : f [[1]] where
   f xss = yss ++ f (xss ++ yss) where
     yss = [y] : map (++ [y]) xss
     y = last (last xss) + 1
-- Reinhard Zumkeller, Nov 16 2013

Maple

T:= proc(n) local i, l, m; l:= NULL; m:= n;
      if n=0 then return 0 fi; for i while m>0 do
      if irem(m, 2, 'm')=1 then l:=l, i fi od; l
    end:
seq(T(n), n=0..50);  # Alois P. Heinz, Sep 06 2014

Mathematica

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 *)
Table[Join@@Position[Reverse[IntegerDigits[n,2]],1],{n,30}] (* Gus Wiseman, Jul 22 2019 *)

Formula

Constructed recursively: subsets that include n are obtained by appending n to all earlier subsets.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2000