A048794 -id:A048794 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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)

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

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

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.

#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
}

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

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

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 *)

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

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

A029931 If 2n = Sum 2^e_i, a(n) = Sum e_i.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 11, 12, 13, 14, 14, 15, 16, 17, 15, 16, 17, 18, 18, 19, 20, 21, 7, 8, 9, 10, 10, 11, 12, 13, 11, 12, 13, 14, 14, 15, 16

Offset: 0

N. J. A. Sloane

Write n in base 2, n = sum b(i)*2^(i-1), then a(n) = sum b(i)*i. - Benoit Cloitre, Jun 09 2002
May be regarded as a triangular array read by rows, giving weighted sum of compositions in standard order. The standard order of compositions is given by A066099. - Franklin T. Adams-Watters, Nov 06 2006
Sum of all positive integer roots m_i of polynomial {m,k} - see link [Shevelev]; see also A264613. - Vladimir Shevelev, Dec 13 2015
Also the sum of binary indices of n, where a binary index of n (A048793) is any position of a 1 in its reversed binary expansion. For example, the binary indices of 11 are {1,2,4}, so a(11) = 7. - Gus Wiseman, May 22 2024

			14 = 8+4+2 so a(7) = 3+2+1 = 6.
Composition number 11 is 2,1,1; 1*2+2*1+3*1 = 7, so a(11) = 7.
The triangle starts:
  0
  1
  2 3
  3 4 5 6
The reversed binary expansion of 18 is (0,1,0,0,1) with 1's at positions {2,5}, so a(18) = 2 + 5 = 7. - _Gus Wiseman_, Jul 22 2019

T. D. Noe, Table of n, a(n) for n = 0..1023
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197, ex. 10. See also DOI.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 3, Theorem 21 and Section 8, Theorem 50)

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A022290 (Fibonacci), A059590 (factorials), A073642, A089625 (primes), A116549, A326031.
Cf. A001793 (row sums), A011782 (row lengths), A059867, A066099, A124757.
Row sums of A048793 and A272020.
Cf. A035327, A056239, A291166, A295235, A326669, A326674, A326675, A326699/A326700.
Contains exactly A000009(n) copies of n.
For length instead of sum we have A000120, complement A023416.
For minimum instead of sum we have A001511, opposite A000012.
For maximum instead of sum we have A029837 or A070939, opposite A070940.
For product instead of sum we have A096111.
The reverse version is A230877, row sums of A371572.
The reverse complement is A359359, row sums of A371571.
The complement is A359400, row sums of A368494.
Numbers k such that a(k) is prime are A372689.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, inverse A048675.
A372471 lists binary indices of primes, row-sums A372429.
Cf. A000009, A030101, A359401, A359402.

a029931 = sum . zipWith (*) [1..] . a030308_row
-- Reinhard Zumkeller, Feb 28 2014

HammingWeight := n -> add(i, i = convert(n, base, 2)):
a := proc(n) option remember; `if`(n = 0, 0,
ifelse(n::even, a(n/2) + HammingWeight(n/2), a(n-1) + 1)) end:
seq(a(n), n = 0..78); # Peter Luschny, Oct 30 2021

a[n_] := (b = IntegerDigits[n, 2]).Reverse @ Range[Length @ b]; Array[a,78,0] (* Jean-François Alcover, Apr 28 2011, after B. Cloitre *)

for(n=0,100,l=length(binary(n)); print1(sum(i=1,l, component(binary(n),i)*(l-i+1)),","))

a(n) = my(b=binary(n)); b*-[-#b..-1]~; \\ Ruud H.G. van Tol, Oct 17 2023

def A029931(n): return sum(i if j == '1' else 0 for i, j in enumerate(bin(n)[:1:-1],1)) # Chai Wah Wu, Dec 20 2022
(C#)
ulong A029931(ulong n) {
    ulong result = 0, counter = 1;
    while(n > 0) {
        if (n % 2 == 1)
          result += counter;
        counter++;
        n /= 2;
    }
    return result;
} // Frank Hollstein, Jan 07 2023

a(n) = a(n - 2^L(n)) + L(n) + 1 [where L(n) = floor(log_2(n)) = A000523(n)] = sum of digits of A048794 [at least for n < 512]. - Henry Bottomley, Mar 09 2001
a(0) = 0, a(2n) = a(n) + e1(n), a(2n+1) = a(2n) + 1, where e1(n) = A000120(n). a(n) = log_2(A029930(n)). - Ralf Stephan, Jun 19 2003
G.f.: (1/(1-x)) * Sum_{k>=0} (k+1)*x^2^k/(1+x^2^k). - Ralf Stephan, Jun 23 2003
a(n) = Sum_{k>=0} A030308(n,k)*A000027(k+1). - Philippe Deléham, Oct 15 2011
a(n) = sum of n-th row of the triangle in A213629. - Reinhard Zumkeller, Jun 17 2012
From Reinhard Zumkeller, Feb 28 2014: (Start)
a(A089633(n)) = n and a(m) != n for m < A089633(n).
a(n) = Sum_{k=1..A070939(n)} k*A030308(n,k-1). (End)
a(n) = A073642(n) + A000120(n). - Peter Kagey, Apr 04 2016

More terms from Erich Friedman

A360287 a(n) is the concatenation of the positions of 1-bits in the binary expansion of the Gray code for n, when 1 is the rightmost position; a(0) = 0.

Original entry on oeis.org

0, 1, 12, 2, 23, 123, 13, 3, 34, 134, 1234, 234, 24, 124, 14, 4, 45, 145, 1245, 245, 2345, 12345, 1345, 345, 35, 135, 1235, 235, 25, 125, 15, 5, 56, 156, 1256, 256, 2356, 12356, 1356, 356, 3456, 13456, 123456, 23456, 2456, 12456, 1456, 456, 46, 146, 1246, 246

Offset: 0

Alois P. Heinz, Feb 01 2023

a(n) represents the n-th finite subset of positive integers in Gray order, two consecutive sets differ in exactly one member: {}, {1}, {1,2}, {2}, {2,3}, {1,2,3}, {1,3}, {3}, {3,4}, {1,3,4}, {1,2,3,4}, {2,3,4}, ... .

a(n) is the concatenation of all terms in the n-th row of A227738 (for n>=1).

			A003188(17) = 25 = 11001_2 gives a(17) = 145.

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Wikipedia, Gray code

Cf. A000225, A000975, A003188, A007908, A048794, A227738.

a:= n-> `if`(n=0, 0, (l-> parse(cat(seq(`if`(l[i]=1, i, [][]),
     i=1..nops(l)))))(Bits[Split](Bits[Xor](n, iquo(n, 2))))):
seq(a(n), n=0..100);

a(2^n-1) = a(A000225(n)) = n.

a(floor(2^(n+1)/3)) = a(A000975(n)) = A007908(n).

A360288 Number T(n,k) of permutations of [n] whose excedance set is the k-th finite subset of positive integers in standard order; triangle T(n,k), n>=0, 0<=k<=ceiling(2^(n-1))-1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 3, 7, 1, 3, 1, 1, 1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1, 1, 31, 15, 115, 7, 69, 31, 115, 3, 37, 17, 69, 7, 37, 15, 31, 1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1, 1, 63, 31, 391, 15, 245, 115, 675, 7, 145, 69

Offset: 0

Alois P. Heinz, Feb 01 2023

The list of finite subsets of positive integers in standard statistical (or Yates) order begins: {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, ... cf. A048793, A048794.
The excedance set of permutation p of [n] is the set of indices i with p(i)>i, a subset of [n-1].
All terms are odd.

			T(5,4) = 3: there are 3 permutations of [5] with excedance set {3} (the 4th subset in standard order): 12435, 12534, 12543.
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  3, 1,  1;
  1,  7, 3,  7, 1,  3, 1,  1;
  1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..15, flattened
Wikipedia, Permutation
Wikipedia, Yates Order

Columns k=0-1 give: A000012, A000225(n-1) for n>=1.
Row sums give A000142.
Row lengths are A011782.
See A152884, A360289 for similar triangles.
Cf. A000027, A029767, A048793, A048794, A263756, A329369.

b:= proc(s, t) option remember; (m->
      `if`(m=0, x^(t/2), add(b(s minus {i}, t+
      `if`(i (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..7);

b[s_, t_] := b[s, t] = Function [m, If[m == 0, x^(t/2), Sum[b[s ~Complement~ {i}, t + If[i < m, 2^i, 0]], {i, s}]]][Length[s]];
T[n_] := CoefficientList[b[Range[n], 0], x];
Table[T[n], {n, 0, 7}]  // Flatten (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

Sum_{k=0..2^(n-1)-1} (k+1) * T(n,k) = A029767(n) for n>=1.

Sum_{k=0..2^(n-1)-1} (2^n-1-k) * T(n,k) = A355258(n+1) for n>=1.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

C

Haskell

Maple

Mathematica

Formula

Extensions

A029931 If 2n = Sum 2^e_i, a(n) = Sum e_i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A360287 a(n) is the concatenation of the positions of 1-bits in the binary expansion of the Gray code for n, when 1 is the rightmost position; a(0) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A360288 Number T(n,k) of permutations of [n] whose excedance set is the k-th finite subset of positive integers in standard order; triangle T(n,k), n>=0, 0<=k<=ceiling(2^(n-1))-1, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula