A344084 -id:A344084 - cp's OEIS frontend

A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 25, 22, 26, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 41, 49, 38, 42, 50, 44, 52, 56, 39, 43, 51, 45, 53, 57, 46, 54, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66, 68, 72, 80, 96, 67, 69, 73, 81, 97, 70, 74, 82, 98, 76, 84, 100, 88, 104, 112, 71, 75, 83

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

Conjecture: For all n, a(A054429(n)) = A054429(a(n)), i.e. A054429 acts as a homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.
From Gus Wiseman, Aug 24 2021: (Start)
As a triangle with row lengths 2^n, T(n,k) for n > 0 appears (verified up to n = 2^15) to be the unique nonnegative integer whose binary indices are the k-th subset of {1..n} containing n. Here, a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion, and sets are sorted first by length, then lexicographically. For example, the triangle begins:
   1
   2  3
   4  5  6  7
   8  9 10 12 11 13 14 15
  16 17 18 20 24 19 21 25 22 26 28 23 27 29 30 31
Mathematica: Table[Total[2^(Append[#,n]-1)]&/@Subsets[Range[n-1]],{n,5}]
Row lengths are A000079 (shifted right). Also Column k = 1.
Row sums are A010036.
Using reverse-lexicographic order gives A059893.
Using lexicographic order gives A059894.
Taking binary indices to prime indices gives A339195 (or A019565).
The ordering of sets is A344084.
A version using Heinz numbers is A344085.
(End)

			From _Gus Wiseman_, Aug 24 2021: (Start)
The terms, their binary expansions, and their binary indices begin:
   0:      ~ {}
   1:    1 ~ {1}
   2:   10 ~ {2}
   3:   11 ~ {1,2}
   4:  100 ~ {3}
   5:  101 ~ {1,3}
   6:  110 ~ {2,3}
   7:  111 ~ {1,2,3}
   8: 1000 ~ {4}
   9: 1001 ~ {1,4}
  10: 1010 ~ {2,4}
  12: 1100 ~ {3,4}
  11: 1011 ~ {1,2,4}
  13: 1101 ~ {1,3,4}
  14: 1110 ~ {2,3,4}
  15: 1111 ~ {1,2,3,4}
(End)

Antti Karttunen, Table of n, a(n) for n = 0..32767
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A209861. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

Cf. A010036, A026793, A048793, A111059, A147655, A246688, A246867, A261144, A272020, A294648, A339360.

a(n) = A209859(A036044(A209641(n))) = A209859(A056539(A209641(n))).

A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 22, 25, 26, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66

Offset: 0

Reinhard Zumkeller, Jan 05 2013

Row lengths are given by Pascal's triangle (cf. A007318), seen as flattened sequence, or for n > 0: length of n-th row = A007318(A003056(n-1),A002262(n-1));
1 <= i < j <= length of n-th row: A023416(T(n,i)) = A023416(T(n,j)), A000120(T(n,i)) = A000120(T(n,j)) and A070939(T(n,i)) = A070939(T(n,j));
the table provides a permutation of the natural numbers when seen as flattened sequence.
This sequence can be seen as an irregular triangle S(i,k) where row 0 = {1}, row n = { m = 2^(n-1)..2^n - 1 } sorted according to omega(A019565(m)), where omega = A001221. Under this arrangement, the rows can be further subdivided into segments of m with the same omega(m), which align with the original definition's triangle T. - Michael De Vlieger, Jan 03 2025

			See link.

Reinhard Zumkeller, Rows n = 0..78 of triangle, flattened - all terms < 2^12
Reinhard Zumkeller, Illustration of initial terms
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^21-1.
Michael De Vlieger, Fan style binary tree of a(n), n = 0..8192, i.e., rows 0..12, with a color function associated with (a(n) mod 2) / 2^floor(log_2 n) that illustrates the relationship with Pascal's triangle.
Index entries for sequences related to binary expansion of n
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences that are permutations of the nonnegative integers

Rows of A187786, duplicates removed;
Cf. A099627 (left edge), A023758 (right edge).
Cf. A019565, A294648, A344084.

import List (elemIndices)
a187769 n k = a187769_tabf !! n !! k
a187769_row n = a187769_tabf !! n
a187769_tabf = [0] : [elemIndices (b, len - b) $
   takeWhile ((<= len) . uncurry (+)) $ zip a000120_list a023416_list |
   len <- [1 ..], b <- [1 .. len]]
a187769_list = concat a187769_tabf

{{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jan 03 2025 *)

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A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

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