A209865 -id:A209865 - 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))).

A209861 Permutation of nonnegative integers which maps A209641 to A209642.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 21, 19, 22, 24, 27, 20, 23, 25, 28, 26, 29, 30, 31, 32, 33, 34, 38, 35, 39, 42, 48, 36, 40, 43, 49, 45, 51, 54, 58, 37, 41, 44, 50, 46, 52, 55, 59, 47, 53, 56, 60, 57, 61, 62, 63, 64, 65, 66, 71, 67, 72, 76, 86, 68, 73, 77, 87, 80, 90, 96, 106, 69, 74, 78, 88, 81, 91, 97, 107, 83

Offset: 0

Antti Karttunen, Mar 24 2012

Conjecture: For all n, A209861(A054429(n)) = A054429(A209861(n)), i.e. A054429 acts as an homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.

The scatterplot graph of the sequence has a nice texture and interesting pattern.

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

Inverse permutation: A209862. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

a(n) = A209640(A209642(n)).

A209863 a(n) = Number of fixed points of permutation A209861/A209862 in range [2^(n-1),(2^n)-1].

Original entry on oeis.org

1, 1, 2, 4, 6, 6, 6, 8, 8, 8, 12, 8, 10, 8, 10, 6, 6, 10, 8, 6, 6, 10, 8, 8, 6

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

See the conjecture given in A209860. If true, then all the terms from a(2) onward are even. a(0) gives the number of fixed points in range [0,0], i.e. 1.

Antti Karttunen, Table of n, a(n) for n = 0..24

A209860, A209864, A209865, A209866, A209867.

A209864 a(n) = number of cycles in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

Original entry on oeis.org

1, 1, 2, 4, 7, 8, 11, 12, 14, 10, 21, 14, 20, 26, 22, 18, 18, 28, 23, 30, 32

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the number of cycles in range [0,0], i.e. 1.

Antti Karttunen, Listing showing a more detailed cycle-structure of permutations A209861/A209862 up to n=20.

Cf. A209863, A209865, A209866, A209867.

A209866 a(n) = least common multiple of all cycle sizes in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 24, 26, 672, 246, 3755388, 13827240, 1768910220, 99034598880, 1463488641762840, 612823600, 171768365608799778, 16338317307187487976, 27491145139913884194480, 14794457633180140325810400, 2084886621890359572790082258379649440

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the LCM of cycle sizes in range [0,0], i.e., 1.

			In range [2^(6-1),(2^6)-1] ([32,63]) of permutations A209861 & A209862, there are 6 cycles of size 1 (fixed points), 2 cycles of size 3, one cycle of size 4, and 2 cycles of size 8 (6 + 2*3 + 4 + 2*8 = 32), thus a(6) = lcm(1,3,4,8) = 24.

Antti Karttunen, Listing showing a more detailed cycle-structure of permutations A209861/A209862 up to n=20.

Cf. A209863, A209864, A209865, A209867.

A209867 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to odd-sized orbits.

Original entry on oeis.org

1, 1, 2, 4, 6, 16, 12, 8, 14, 8, 406, 8, 56, 80, 1686, 8866, 8272, 15178, 9462, 938, 41128

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the number of odd sized cycles in range [0,0], i.e. 1, as there is just one fixed point in that range.

			In range [2^(6-1),(2^6)-1] ([32,63]) of permutations A209861 & A209862, there are 6 cycles of size 1 (six fixed points), 2 cycles of size 3, one cycle of size 4, and 2 cycles of size 8, 6*1 + 2*3 + 1*4 + 2*8 = 32 in total, of which 6*1 + 2*3 elements are in odd-sized cycles, thus a(6)=12.

a(n) = A000079(n-1) - A209868(n) for all n>0. Cf. A209860, A209863, A209864, A209865, A209866.

A209868 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to even-sized orbits.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 20, 56, 114, 248, 106, 1016, 1992, 4016, 6506, 7518, 24496, 50358, 121610, 261206, 483160

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the number of even-sized cycles in range [0,0], i.e. 0, as there is only one fixed point in that range.

			In range [2^(6-1),(2^6)-1] ([32,63]) of permutations A209861 & A209862, there are 6 cycles of size 1 (six fixed points), 2 cycles of size 3, one cycle of size 4, and 2 cycles of size 8, i.e. 6*1 + 2*3 + 1*4 + 2*8 = 32 in total, of which 4 + 2*8 elements are in even-sized cycles, thus a(6)=20.

a(n) = A000079(n-1) - A209867(n) for all n>0. Cf. A209860, A209863, A209864, A209865, A209866.

cp's OEIS Frontend

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

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A209861 Permutation of nonnegative integers which maps A209641 to A209642.

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A209863 a(n) = Number of fixed points of permutation A209861/A209862 in range [2^(n-1),(2^n)-1].

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A209864 a(n) = number of cycles in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

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A209866 a(n) = least common multiple of all cycle sizes in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

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A209867 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to odd-sized orbits.

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A209868 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to even-sized orbits.

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