User:Yosu Yurramendi/Q+

Enumeration systems of positive rationals ( ℚ⁺ )
based on Stern's sequence ( A002487 )

Introduction and more detailed contents

A002487 can be represented by blocks of 2^m terms (m ≥ 0) in a natural way:   1  1 2  1 3 2 3  1 4 3 5 2 5 3 4 ... ([1], [2]). This representation is basic for this work. Here is a spiral representation: [3]
In order to compute the value of its n^th term, beyond the recursive formula of its definition, a fast algorithm for A002487 has been defined: ∀n > 0, it computes A002487(n) by taking into account the binary representation of n ([4], translator: [5]).

The main aim of this work is just to go in depth to the structure of a set of some well-known enumeration systems of ℚ⁺ based on sequence A002487. See 'Index entries for fraction trees' ([6]).

• The well-known enumeration systems has been grouped in two classes by means of a combinatorial procedure by taking into account the terms of the first three blocks of A002487:
• m = 0. {1}:    1 / 1.
• m = 1. {1, 2}:    1 / 2,  2 / 1.
• m = 2. {1, 2, 3, 3}: [7],  [8].
• More classes could be defined ([9], [10], translator: [11]).
• Definitions and generation algorithms are given at sections 2.1 and 2.2.
• Graphical representation of ℚ⁺: [12]. An enumeration system is represented by a trajectory that passes through each rational point (red colour) only once. Thus, ℚ⁺ is covered. Coverage based on the Stern sequence is done in blocks: [13].
• The infinite sequence A002487(n)/A002487(1+n), n > 0, constitutes an enumeration system of ℚ⁺ ([14]), and it belongs to Class 1:
    1  1 2  1 3 2 3  1 4 3 5 2 5 3 4  1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 ...
    1  2 1  3 2 3 1  4 3 5 2 5 3 4 1  5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 ...
  The beginning of its trajectory in the plane of ℚ⁺: [15]
• Let σ be an infinite permutation (bijection) on ℕ such that if 2^m ≤ n < 2^m+1 then 2^m ≤ σ(n) < 2^m+1. That is to say, it is itself a composition of finite-range permutations, where everyone works within an interval [2^m, ..., 2^m+1) (m ≥ 0), which is composed by 2^m integer terms.
  • ∀m ≥ 0, ∀n > 0 such that n < 2^m+1 σ^(2m)(n) = n.
  • If σ(2) = 2 and σ(3) = 3 (alternatively, σ(2) = 3 and σ(3) = 2), then A002487∘σ / A002487∘(1+σ) (alternatively, A002487∘(1+σ) / A002487∘σ) constitutes an enumeration system of ℚ⁺.
  • In this sense each considered well-known enumeration system could be characterized by A002487 and a particular permutation σ. See 3.2 which they are. Thus, every numerator (denominator) is a permuted sequence of A002487 in each block of 2^m terms (m > 0).
• A very desirable property of an enumeration system of ℚ⁺, further than the existence of a bijection or an one-to-one correspondence with ℕ, is to have some procedures (algorithms) which can specify (as fast as possible) its nature:
  • What positive rational corresponds to a given location (see 2.1.2 and 2.2.2).
  • The fast algorithm for A002487 beside a fast algorithm for the particular permutation σ of an enumeration system give the positive rational corresponding to a given location.
  • And, viceversa, what location corresponds to a given positive rational (see 2.1.3 and 2.2.3).
• Permutations between numerator and denominator of each system (see 3.1) as well as permutations between systems have been pointed out (within each class, see 3.3.1, and between classes, 3.3.2).
• Sequences num+den (= numerator+denominator) are also considered.
  • numerator/num+den and denominator/num+den are enumeration systems of rationals in (0,1).
  • numerator and denominator can be computed from num+den (see 2.1.4 and 2.2.4).
  • num+den sequences can be computed as permutations of A007306.
  • A007306 is the num+den of two well-known enumeration systems, one of each class (see 3.3.1).
  • A fast algorithm for A007306 has also been defined, in order to compute the value of its n^th term: [16]. It is very similar to that of A002487.
• Section 4 deals with the frequencies of each integer in each block of A002487 and A007306. This issue concerns all enumeration systems based on A002487 and a σ permutation defined as above.

Classification

Class 1

Four enumeration systems: f_CW, f_DRIB, f_YT, f_YU-1 (f_* can be any of them):

                   numerator_*(n)                          num+den_*(n)
∀n > 0    f_*(n) = ---------------             f_*(n)+1 = ---------------
                  denominator_*(n)                       denominator_*(n)

 

• Calkin-Wilf ([17]):     f_CW  (2n  ) =  f_CW  (n)    / (f_CW  (n)+1) = numerator_CW    (n) / num+den_CW       (n)
                   f_CW  (2n+1) = (f_CW  (n)+1) /       1  = num+den_CW       (n) / denominator_CW  (n)
• driB ([18]):         f_DRIB(2n  ) =        1  / (f_DRIB(n)+1) = denominator_DRIB(n) / num+den_DRIB     (n)
                   f_DRIB(2n+1) = (f_DRIB(n)+1) /  f_DRIB(n)    = num+den_DRIB     (n) / numerator_DRIB  (n)
• Yu-Ting^-1 ([19], [20]): f_YT  (2n  ) =  f_YT  (n)    / (f_YT  (n)+1) = numerator_YT     (n) / num+den_YT      (n)
                   f_YT  (2n+1) = (f_YT  (n)+1) /  f_YT  (n)    = num+den_YT       (n) / numerator_YT    (n)
  
• Yurramendi-1:         f_YU-1(2n  ) =       1  / (f_YU-1(n)+1) = denominator_YU-1(n) / num+den_YU-1     (n)
                   f_YU-1(2n+1) = (f_YU-1(n)+1) /       1   = num+den_YU-1    (n) / denominator_YU-1(n)

 

The four systems organized in a table

f_*(1) = 1/1, ∀n > 0   

                 f_*(2n) =    f_*(n)/(f_*(n)+ 1)             1/(f_*(n)+1)
                           --------------------------------------------------
            (f_*(n)+1)/1    |  Calkin-Wilf (f_*≡f_CW)    Yurramendi-1 (f_*≡f_YU-1) |
f_*(2n+1) =                 |                                                |
           (f_*(n)+1)/f_*(n) |   Yu-Ting^-1(f_*≡f_YT)         driB (f_*≡f_DRIB)      |
                           --------------------------------------------------

For all four systems:    ∀n > 0    f_*(2n) < 1, f_*(2n+1) > 1.

Relation between a rational and its inverse:   

∀n > 0  or  ∀m ≥ 0, ∀k | 0 ≤ k < 2^m   (n ≡ 2^m + k)
                 f_CW  (2^m+k) ·f_CW   (2^m+1-1-k)  = 1
                 f_DRIB(2^m+k) ·f_DRIB (2^m+1-1-k)  = 1
                 f_YT  (2n)   ·f_YT   (2n+1)     = 1 
                 f_YU-1(2n)   ·f_YU-1 (2n+1)     = 1

The numerator and denominator of all four systems can be also organized in a table

numerator_*(1) = 1, denominator_*(1) = 1, ∀n > 0   

                    numerator_*(2n) =      numerator_*(n)       denominator_*(n)
                                    -------------------------------------------
                    denominator_*(n) |     Calkin-Wilf         Yurramendi-1    |
denominator_*(2n+1) =                |                                         |
                      numerator_*(n) |       Yu-Ting^-1           driB          |
                                    -------------------------------------------

numerator_*(2n+1) = denominator_*(2n) = num+den_*(n).

 

Generation of the four enumeration systems

R code ([21])

• By fractions: [22].
• By numerator and denominator: [23]. Video: [24]
• By continued fractions: [25].

Graphic representation of enumeration systems in the plane of ℚ⁺: [26] .

Given a system and a location, compute its positive rational

• By continued fractions: [27]

Given a system and a positive rational, compute its location

• By a procedure based on an Euclidean algorithm ([28]): [29]
• By continued fractions: [30]

Sequences in the OEIS

• Calkin-Wilf:  A002487 / A002487' ,   A007306, where A002487'(n) = A002487(n+1), n > 0
• driB:         A162911 / A162912  ,  A268087
• Yu-Ting^-1:    A020651 / A020650  ,  A086592
• Yurramendi-1: A245327 / A245328  ,   A273493

The relationships between numerator, denominator, and num+den are more general than the previous equalities numerator(2n+1) = denominator(2n) = num+den(n). (It is the case q = 0)(see R code [31] for involved relations):

• Calkin-Wilf  : numerator(2^q*(2n+1)    ) = denominator(2^q*(2n+1) - 1) = num+den(n), n > 0, q ≥ 0.
                 A002487  = A002487'∘A153151 ,  A002487' = A002487∘A153152,  A153151∘A153152 = A153152∘A153151 = A000027.  (implicit relationship).
                 numerator(2^q           ) = denominator(2^q*2      - 1) = 1,            q ≥ 0.
                 A002487∘A000079          = A002487'∘A000225           = 1
• driB         : numerator(2^q*(2n+1) - (-2)^q*2/3                   - 1/3) = denominator(2^q*(2n+1) + (-2)^q*2/3                   - 2/3) = num+den(n), n > 0, q ≥ 0.
                 A162911 = A162912∘A334998 ,    A162912 = A162911∘A334999 ,  A334998∘A334999 = A334999∘A334998 = A000027 .
                 numerator(2^q        - (-2)^q*1/3 - [2^q-2 - (-2)^q-2] - 2/3) = denominator(2^q        + (-2)^q*1/3 - [2^q-2 + (-2)^q-2] - 1/3)        = 1,q ≥ 0.
                 A162911∘A061547                                          = A162912∘A086893 .                                          = 1
• Yu-Ting      : numerator(2^q*(2n+1)    ) = denominator(    2n       ) = num+den(n), n > 0, q = 0.
                 numerator(2^q*(2n+1)    ) = denominator(2^q*(2n+1) + 1) = num+den(n), n > 0, q > 0.
                 A020651 = A020650∘A065190 ,    A020650 = A020651∘A065190, A065190 ∘A065190 = A000027 .
                 numerator(2^q           ) = denominator(2^q         + 1) = 1,                 q > 0. numerator(1) = denominator(1) = 1.
                 A020651∘A000079          = A020650∘A083318            = 1
• Yurramendi-1 : numerator(   (2n+1)    ) = denominator(2^q*(2n+1) - 1) = num+den(n), n > 0, q = 0.
                 numerator(2^q*(2n+1) - 2) = denominator(2^q*(2n+1) - 1) = num+den(n), n > 0, q > 0.
                 A245327 = A245328∘A065190 ,    A245328 = A245327∘A065190, A065190 ∘A065190 = A000027.
                 numerator(2^q*2      - 2) = denominator(2^q*2      - 1) = 1,             q > 0. numerator(1) = denominator(1) = 1.
                 A245327∘A095121          = A245328∘A000225            = 1

Class 2

Four enumeration systems: f_SB, f_BIRD, f_HCS, f_YU-2 (f_* can be any of them):

 ∀m ≥ 0, ∀k such that 0 ≤ k < 2^m  

            numerator_*(2^m+k)                           num+den_*(2^m+k)
f_*(2^m+k) = ------------------           f_*(2^m+k)+1 = ------------------
           denominator_*(2^m+k)                        denominator_*(2^m+k)

 

• Stern-Brocot ([32]): f_SB  (2^m+1   +k) =  f_SB  (2^m+k)    / (f_SB  (2^m+k)+1) = numerator_SB     (2^m+k) / num+den_SB       (2^m+k)
                   f_SB  (2^m+1+2^m+k) = (f_SB  (2^m+k)+1) /             1  = num+den_SB      (2^m+k) / denominator_SB  (2^m+k)
• Bird ([33]):         f_BIRD(2^m+1   +k) =              1  / (f_BIRD(2^m+k)+1) = denominator_BIRD(2^m+k) / num+den_BIRD     (2^m+k)
                   f_BIRD(2^m+1+2^m+k) = (f_BIRD(2^m+k)+1) /  f_BIRD(2^m+k)     = num+den_BIRD     (2^m+k) / numerator_BIRD  (2^m+k)
• HCS ([34]):          f_HCS (2^m+1   +k) =  f_HCS (2^m+k)    / (f_HCS (2^m+k)+1)  = numerator_HCS   (2^m+k) / num+den_HCS      (2^m+k)
                    f_HCS (2^m+1+2^m+k) = (f_HCS (2^m+k)+1) /  f_HCS (2^m+k)     = num+den_HCS     (2^m+k) / numerator_HCS    (2^m+k)
• Yurramendi-2:          f_YU-2(2^m+1   +k) =             1  / (f_YU-2(2^m+k)+1)  = denominator_YU-2(2^m+k) / num+den_YU-2     (2^m+k)
                   f_YU-2(2^m+1+2^m+k) = (f_YU-2(2^m+k)+1) /             1   = num+den_YU-2     (2^m+k) / denominator_YU-2(2^m+k)

 

The numerator and denominator of four systems organized in a table

numerator_*(1) = 1, denominator_*(1) = 1. ∀m ≥ 0, ∀k such that 0 ≤ k < 2^m

                   numerator_*(2^m+1+k) =         numerator_*(2^m+k)  denominator_*(2^m+k)
                                            -----------------------------------------
                         denominator_*(2^m+k) |    Stern-Brocot       Yurramendi-2    |
denominator_*(2^m+1+2^m+k) =                    |                                       |
                           numerator_*(2^m+k) |       HCS                Bird         |
                                            -----------------------------------------

numerator_*(2^m+1+2^m+k) = denominator_*(2^m+1+k) = num+den_*(2^m+k).

 

Generation of the four enumeration systems

• By fractions: [35].
• By numerator and denominator: [36]. Video: [37]
• By continued fractions: [38].

Graphic representation of enumeration systems in the plane of ℚ⁺: [39] .

Given a system and a location, compute its positive rational

• By continued fractions: [40]

Given a system and a positive rational, compute its location

• By a procedure based on an Euclidean algorithm ([41]): [42]
• By continued fractions: [43]

Sequences in the OEIS

• Stern-Brocot: A007305 / A047679 ,   A007306
• Bird        : A162909 / A162910 ,   A268087
• HCS         : A071766 / A229742 ,   A071585
• Yurramendi-2: A245325 / A245326 ,   A273494

The relationships between numerator, denominator, and num+den are more general than the previous equalities numerator(2^m+1+2^m+k) = denominator(2^m+1+k) = num+den(2^m+k).(it is the case q = 1)(see R code [44] for involved relations):

• Stern-Brocot : numerator((  2^q + 1)*2^m + k) = denominator((  2^q - 1)*2*2^m + k) = num+den(2^m + k), m ≥ 0, 0 ≤ k < 2^m, q > 0.
                 A007305 = A047679∘A153141 ,    A047679 = A007305∘A153142,  A153141∘A153142 = A153142∘A153141 = A000027.   (implicit relationship).
                 numerator(   2^q-1          ) = denominator(   2^q-1    *2    - 1) = 1,                                 q > 0.
                 A007305∘A000079              = A047679∘A000225                  = 1
• Bird         : numerator((8*2^q - 3 - 5*(-1)^q)*1/6*2^m + k) = denominator((5*2^q - 3 - 5*(-1)^q)*1/6*2^m + k) = num+den(2^m + k), m ≥ 0, 0 ≤ k < 2^m, q > 0.
                 A162909 = A162910∘A154448 ,                  A162910 = A162909∘A154447,  A154448∘A154447 = A154447∘A154448 = A000027.
                 numerator((4*2^q - 3 -   (-1)^q)*1/6)        = denominator( 5*2^q - 3 +   (-1)^q)*1/6)        = 1, q > 0.
                 A162909∘A000975                            = A162910∘A081254 .                              = 1
• HCS          : numerator((  2^q + 1)*2^m + k) = denominator(           2*2^m + k) = num+den(2^m + k), m ≥ 0, 0 ≤ k < 2^m, q = 1.
                 numerator((  2^q + 1)*2^m + k) = denominator((3*2^q + 1)*  2^m + k) = num+den(2^m + k), m ≥ 0, 0 ≤ k < 2^m, q > 1.
                 A071766 = A229742∘A063946 ,    A229742 = A071766∘A063946, A063946 ∘A063946 = A000027.
                 numerator(   2^q            ) = denominator( 3*2^q-1            ) = 1,                                  q > 0. numerator(1) = denominator(1) = 1.
                 A071766∘A000079              = A229742∘A003945                 = 1
• Yurramendi-2 : numerator(         3*2^m + k) = denominator((  2^q - 1)*2*2^m + k) = num+den(2^m + k), m ≥ 0, 0 ≤ k < 2^m, q = 1.
                 numerator((3*2^q - 2)*2^m + k) = denominator((  2^q - 1)*2*2^m + k) = num+den(2^m + k), m ≥ 0, 0 ≤ k < 2^m, q > 1.
                 A245325 = A245326∘A063946 ,    A245326 = A245325∘A063946, A063946 ∘A063946 = A000027.
                 numerator( 3*2^q-1 - 1      ) = denominator(   2^q     *2    - 1) = 1,                                  q > 0. numerator(1) = denominator(1) = 1.
                 A245325∘A083329              = A245326∘A000225                 = 1

Other classes

Other classes could be defined. For example, in a figure given at the beginning of this webpage ([45]) the gaps shown at the bottom right could be fulfilled by the following class (four systems). Video: [46].
The same way to relate Class 1 and Class 2 (A059893, the bit-reversal permutation by blocks of 2^m terms; see permutations 'Between Class 1 and Class 2') could serve to define a new class. This one fulfills also those gaps.
Stern 's sequence based enumeration systems are not exhaustive given the examples of all permutations of m = 2 (m = 0 and m = 1 are fixed). These procedures are quite arbitrary. It is possible for a system to have the same fractions for m ≤ 2 as one of the above systems, and different order of fractions in other blocks. For example, system A002487∘(1+A006068) / A002487∘A006068 ≡ A324338 / A324337 has the same permutation (m = 2) as the upper left of unknown gaps, A002487∘A122155 / A002487∘(1+A122155) has the same beginning (m ≤ 2) Yu-Ting system of Class 1, and A002487∘A122155∘A059893 / A002487∘(1+A122155∘A059893) the same (m ≤ 2) as HCS from Class 2.

Permutations

Some useful general permutations of ℕ:
ϵ ≡ A000027, the positive integers (identity permutation).
η ≡ A054429, the inverse permutation by blocks of 2^m terms.
τ₁ ≡ A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1), n > 0.
τ₂ ≡ A063946: 1 is fixed, followed by infinite number of adjacent transpositions (2^m+1+k, 2^m+1+2^m+k), m ≥ 0, 0 ≤ k < 2^m.
μ₁ ≡ A092569: Permutation of integers a(a(n)) = n. In binary representation of n, transformation of inner bits, 1 <-> 0, gives binary representation of a(n).
μ₂ ≡ A117120: a(1)=1. a(n) is smallest positive integer not occurring earlier in the sequence where a(n) is congruent to -1 (mod a(n-1)).

• η∘η = ϵ.
• ({ϵ, η}, ∘) is a group (C2, [47]).

• τ₁∘τ₁ = τ₂∘τ₂ = ϵ.
• μ₁∘μ₁ = μ₂∘μ₂ = ϵ.
• For η, τ₁, τ₂, μ₁, μ₂, ∀m > 1, there are 2^m-1 orbits of length 2. For m = 0 one fixed point, and for m = 1 two fixed points. Therefore, they all are self-inverse.

• η∘τ₁ = τ₁∘η = μ₁, η∘τ₂ = τ₂∘η = μ₂.
• ({ϵ, η, τ₁, μ₁}, ∘) is a Klein 4-group (C2xC2, [48]), as well as ({ϵ, η, τ₂, μ₂}, ∘).

• τ₁∘τ₂ = τ₂∘τ₁ = μ₁∘μ₂ = μ₂∘μ₁.
• ({ϵ, η, τ₁, τ₂, μ₁, μ₂, τ₁∘τ₂, η∘τ₁∘τ₂}, ∘) is an elementary abelian group of order 2³ (C2xC2xC2, [49]).

π₁ ≡ A059893, the bit-reversal permutation by blocks of 2^m terms.
π₂ ≡ A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.

• π₁∘π₁ = π₂∘π₂ = ϵ.
  π₁∘π₂ = π₂∘π₁ = η, π₁∘η = η∘π₁ = π₂, π₂∘η = η∘π₂ = π₁.
  ({ϵ, η, π₁, π₂}, ∘) is a Klein 4-group (C2xC2, [50]).
• For π₁, π₂ ∀m > 1, there are A016116(m+1) (= 2^{⌊(m+1)/2)⌋}) fixed points, and A032085(m) orbits of length 2. There are altogether A005418(m+1) orbits. They both are self-inverse.

• π₁∘τ₁ = τ₂∘π₁ , τ₁∘π₁ = π₁∘τ₂
  π₂∘τ₁ = τ₂∘π₂ , τ₁∘π₂ = π₂∘τ₂
  π₁∘μ₁ = μ₂∘π₁ , μ₁∘π₁ = π₁∘μ₂
  π₂∘μ₁ = μ₂∘π₂ , μ₁∘π₂ = π₂∘μ₂

Between numerator and denominator of systems

Class 1

η ≡ A054429, the inverse permutation by blocks of 2^m terms.
τ₁ ≡ A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).

• Calkin-Wilf:              A002487∘η  = A002487', A002487'∘η = A002487
                                                   A002487∘A153152  =   A002487',      A002487'∘A153151 =   A002487  ,  A153151∘A153152 = A153152∘A153151 = A000027 
• driB:                     A162911∘η  = A162912,  A162912∘η  = A162911
                                                   A162911∘A334999  =   A162912,       A162912∘A334998  =   A162911  ,  A334998∘A334999 = A334999∘A334998 = A000027 
• Yu-Ting:                  A020651∘τ₁ = A020650,  A020650∘τ₁ = A020651
• Yurramendi-1:             A245327∘τ₁ = A245328,  A245328∘τ₁ = A245327
• Yu-Ting and Yurramendi-1: A020651∘η  = A245327,  A245327∘η  = A020651
                            A245328∘η  = A020650,  A020650∘η  = A245328

Class 2

η ≡ A054429, the inverse permutation by blocks of 2^m terms.
τ₂ ≡ A063946: write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.

• Stern-Brocot:             A007305∘η  = A047679,  A047679∘η  = A007305 (by coordinating carefully the n^th term)
                                             A007305∘A153142  =   A047679           A047679∘A153141  =   A007305,   A153141∘A153142 = A153142∘A153141 = A000027 

• Bird:                     A162909∘η  = A162910,  A162910∘η  = A162909
                                             A162909∘A154447  =   A162910,         A162910∘A154448  =   A162909,  A154448∘A154447 = A154447∘A154448 = A000027
• HCS:                      A071766∘τ₂ = A229742,  A229742∘τ₂ = A071766
• Yurramendi-2:             A245325∘τ₂ = A245326,  A245326∘τ₂ = A245325
• HCS and Yurramendi-2  :   A071766∘η  = A245326,  A245326∘η  = A071766
                            A245325∘η  = A229742,  A229742∘η  = A245325

Between A002487 and numerator (denominator) of systems

All these numerators and denominators can be obtained through A002487 and some permutations of ℕ which can be quickly computed by copying and pasting the following codes (given n, the permutation is computed taking into account the binary representation of n): [51].

Class 1

ϵ ≡ A000027, the positive integers (identity permutation).
η ≡ A054429, the inverse permutation by blocks of 2^m terms.
τ₁ ≡ A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).
μ₁ ≡ A092569: Permutation of integers a(a(n)) = n. In binary representation of n, transformation of inner bits, 1 <-> 0, gives binary representation of a(n).
π₁ ≡ A059893, the bit-reversal permutation by blocks of 2^m terms.
π₂ ≡ A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.

• Calkin-Wilf:   A002487∘ϵ         / A002487∘(1+ϵ)     = A002487 / A002487'
                  A002487∘(1+η∘ϵ)     / A002487∘η∘ϵ      = A002487 / A002487'
• driB:           A002487∘   A258996 / A002487∘(1+A258996) = A162911 / A162912
                  A002487∘(1+η∘A258996) / A002487∘η∘ A258996  = A162911 / A162912
                  A002487∘(1+ A332769) / A002487∘  A332769  = A162911 / A162912
                  # ({ϵ, η, τ₁, μ₁,  A258996,  η∘ A258996, τ₁∘A258996, μ₁∘A258996}, ∘)
                  # ({ϵ, η, τ₁, μ₁, η∘A332769,   A332769, μ₁∘A332769, τ₁∘A332769}, ∘)
                  # ({ϵ, η, τ₁, μ₁, μ₁∘A284447, τ₁∘A284447, η∘A284447,    A284447}, ∘)
                  # they all three are the same elementary abelian group of order 2³ (C2xC2xC2, [52]).
                  # A002487∘A284447 / A002487∘(1+A284447) is not 'driB' system.
                  # μ₁ = A258996∘A284447 = A284447∘A258996,
                  # For A258996, A332769, A284447, ∀m > 1, there are 2^m-1 orbits of length 2. Therefore, they all three are self-inverse.
                  # ∀m ≥ 0, ∀k 0 ≤ k < 2^m, if m odd  π₁∘A258996(2^m+k) = A258996∘π₁(2^m+k),
                              if m even π₁∘A258996(2^m+k) = A258996∘π₁(2^m+1-1-k)
                  # ∀m ≥ 0, ∀k 0 ≤ k < 2^m, if m odd  π₂∘A258996(2^m+k) = A258996∘π₂(2^m+k),
                              if m even π₂∘A258996(2^m+k) = A258996∘π₂(2^m+1-1-k)
  
• Yu-Ting:        A002487∘   A231551 / A002487∘(1+A231551) = A020651 / A020650
                  A002487∘(1+η∘A231551) / A002487∘η∘ A231551  = A020651 / A020650
                  A002487∘(1+ A153154) / A002487∘  A153154  = A020651 / A020650
                  # For A231551
                     ∀m > 0, there are 2 fixed points in positions 2^m and 2^m+2^m-1,
                     ∀m > 1, there is an unique orbit of length 2, in positions 2^m+2^m-2 and 2^m+1-2^m-2,
                     ∀m > 2 (m = 2^q + r, 1 ≤ r ≤ 2^q) and ∀p > 0 such that 2^p is the length of a orbit,
                     the number of orbits t(m, p) is given by the following formula:
                     t(m,p) = 0                  if p > (q+1)
                            = 2^A000295(p-1)*(2^r   -1) if p = (q+1)
                            = 2^A000295(p-1)*(2^2(p-1)-1) if p < (q+1)
                     ∀m ≥ 0 there are altogether A007886(m) orbits.
                     ∀m ≥ 0, ∀n > 0 such that n ≤ 2^m+2 + 2^m+1 (higher than 2^m), one has A231551^(2m)(n) = n.
                  # For A153154
                     ∀m > 1 (m = 2^q + r, 0 ≤ r < 2^q), there are 2^{(A000295(q)+r)} orbits of length 2^q+1.
                     ∀m > 0 there are A054243(m) orbits.
                     ∀m ≥ 0, ∀n > 0 such that n ≤ 2^m, one has A153154^(2m)(n) = n.
  
• Yurramendi-1:   A002487∘   A284459 / A002487∘(1+A284459) = A245327 / A245328
                  A002487∘(1+η∘A284459) / A002487∘η∘ A284459  = A245327 / A245328
                  A002487∘(1+ A154437) / A002487∘  A154437  = A245327 / A245328
                  # For A284459 the orbit analysis is the same as that of A231551.
                  # For A154437 the orbit analysis is the same as that of A153154.

Relationships between these permutations:

• _

  _______

  _

  _______

  _______

  _________

  _______

  __________

  __________

  _______________

  _______________

  ________________

  _______________

  _______________

  _______________

  _

  |

  ∘

  |

  ϵ

  η

  τ1

  μ1

  π1

  π2

  A258996

  A332769

  A231551

  A153154

  A284459

  A154437

  |

  _

  _______

  _

  _______

  _______

  _________

  _______

  __________

  __________

  _______________

  _______________

  ________________

  _______________

  _______________

  _______________

  _

  |

  ϵ

  |

  ϵ

  η

  τ1

  μ1

  π1

  π2

  A258996

  A332769

  A231551

  A153154

  A284459

  A154437

  |

  |

  η

  |

  η

  ϵ

  μ1

  τ1

  π2

  π1

  A332769

  A258996

  A153154

  A231551

  A154437

  A284459

  |

  |

  τ1

  |

  τ1

  μ1

  ϵ

  η

  π1∘τ2

  π2∘τ2

  η∘A284447

  A284447

  τ1∘A231551

  τ1∘A153154

  τ1∘A284459

  τ1∘A154437

  |

  |

  μ1

  |

  μ1

  τ1

  η

  ϵ

  μ1∘π1

  μ1∘π2

  A284447

  A332769∘μ1

  μ1∘A231551

  μ1∘A153154

  μ1∘A284459

  μ1∘A154437

  |

  |

  π1

  |

  π1

  π2

  τ2∘π1

  π1∘μ1

  ϵ

  η

  π2∘A332769

  π2∘A258996

  π2∘A153154

  π2∘A231551

  π2∘A154437

  π2∘A284459

  |

  |

  π2

  |

  π2

  π1

  τ2∘π2

  π2∘μ1

  η

  ϵ

  π1∘A332769

  π1∘A258996

  π1∘A153154

  π1∘A231551

  π1∘A154437

  π1∘A284459

  |

  |

  A258996

  |

  A258996

  A332769

  η∘A284447

  A284447

  A332769∘π2

  A332769∘π1

  ϵ

  η

  A284459

  A154437

  A231551

  A153154

  |

  |

  A332769

  |

  A332769

  A258996

  A284447

  μ1∘A332769

  A258996∘π2

  A258996∘π1

  η

  ϵ

  A154437

  A284459

  A153154

  A231551

  |

  |

  A231551

  |

  A231551

  A154437

  A153154

  A284459

  A154437∘π2

  A154437∘π1

  A154437∘A332769

  A154437∘A258996

  A154437∘A153154

  A154437∘A231551

  A154437∘A154437

  A154437∘A284459

  |

  |

  A153154

  |

  A153154

  A284459

  A231551

  A154437

  A284459∘π2

  A284459∘π1

  A284459∘A332769

  A284459∘A258996

  A284459∘A153154

  A284459∘A231551

  A284459∘A154437

  A284459∘A284459

  |

  |

  A284459

  |

  A284459

  A153154

  A154437

  A231551

  A153154∘π2

  A153154∘π1

  A153154∘A332769

  A153154∘A258996

  A153154∘A153154

  A153154∘A231551

  A153154∘A154437

  A153154∘A284459

  |

  |

  A154437

  |

  A154437

  A231551

  A284459

  A153154

  A231551∘π2

  A231551∘π1

  A231551∘A332769

  A231551∘A258996

  A231551∘A153154

  A231551∘A231551

  A231551∘A154437

  A231551∘A284459

  |

  _

  _______

  _

  _______

  _______

  _________

  ________

  __________

  __________

  _______________

  _______________

  ________________

  _______________

  _______________

  _______________

  _

• A258996∘A258996 = ϵ
  A258996∘A231551 = A284459
  A258996∘A284459 = A231551
• A284459 = η∘A231551∘η, A231551 = η∘A284459∘η
  A154437 = η∘A153154∘η, A153154 = η∘A154437∘η
• A231551∘A258996 = A154437∘A332769 = A284459∘A284447
  A153154∘A258996 = A284459∘A332769 = A154437∘A284447
  A284459∘A258996 = A153154∘A332769 = A231551∘A284447
  A154437∘A258996 = A231551∘A332769 = A153154∘A284447

<p>Inversely, A002487 can be obtained through all of these numerators and denominators and some permutations of ℕ, which are just the inverse of the permutations above. That is to say:

• A002487 = A162911∘A258996 = A020651∘A231550 = A245327∘A284460.
• A002487 = A162912∘A332769 = A020650∘A153153 = A245328∘A154438.

Class 2

ϵ  ≡ A000027, the positive integers (identity permutation).
η  ≡ A054429, the inverse permutation by blocks of 2^m terms.
τ₂ ≡ A063946: write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.
μ₂ ≡ A117120: a(1)=1. a(n) is smallest positive integer not occurring earlier in the sequence where a(n) is congruent to -1 (mod a(n-1)).
π₁ ≡ A059893, the bit-reversal permutation by blocks of 2^m terms.
π₂ ≡ A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.

• Stern-Brocot:   A002487∘π₁∘ϵ       / A002487∘(1+π₁∘ϵ)     = A007305 / A047679
                  A002487∘(1+π₂∘ϵ)    / A002487∘π₂∘ϵ         = A007305 / A047679
                  A002487∘(1+π₁∘η)    / A002487∘π₁∘η         = A007305 / A047679
                  A002487∘π₂∘η       / A002487∘(1+π₂∘η)     = A007305 / A047679
• Bird:           A002487∘π₁∘   A258746 / A002487∘(1+π₁∘A258746) = A162909 / A162910
                  A002487∘(1+π₂∘A258746) / A002487∘π₂∘   A258746 = A162909 / A162910
                  A002487∘(1+π₁∘A165199) / A002487∘π₁∘   A165199 = A162909 / A162910
                  A002487∘π₂∘   A165199 / A002487∘(1+π₂∘A165199) = A162909 / A162910
                  # ({ϵ, η, τ₂, μ₂,  A258746,  η∘A258746, τ₂∘A258746, μ₂∘A258746}, ∘)
                  # ({ϵ, η, τ₂, μ₂, η∘ A165199,  A165199, μ₂∘A165199, τ₂∘A165199}, ∘)
                  # ({ϵ, η, τ₂, μ₂, μ₂∘A284120, τ₂∘A284120, η∘A284120,    A284120}, ∘)
                  # they all three are the same elementary abelian group of order 2³ (C2xC2xC2, [53]).
                  # A002487∘A284120 / A002487∘(1+A284120) is not a 'Bird' system.
                  # μ₂ = A258746∘A284120 = A284120∘A258746.
                  # For A258746, A165199, A284120, ∀m > 1, there are 2^m-1 orbits of length 2. For m = 0 one fixed point, and for m = 1 two fixed points. Therefore, they all are self-inverse.
                  # ∀m ≥ 0, ∀k 0 ≤ k < 2^m, if m odd  π₁∘A258746(2^m+k) = A258746∘π₁(2^m+k),
                              if m even π₁∘A258746(2^m+k) = A258746∘π₁(2^m+1-1-k)
                  # ∀m ≥ 0, ∀k 0 ≤ k < 2^m, if m odd  π₂∘A258746(2^m+k) = A258746∘π₂(2^m+k),
                              if m even π₂∘A258746(2^m+k) = A258746∘π₂(2^m+1-1-k)
                  # ∀m ≥ 0, ∀k 0 ≤ k < 2^m, if m odd  A258746(2^m+k) = A258996(2^m+k),
                              if m even A258746(2^m+k) = A258996(2^m+1-1-k)
                  #   A258996∘π₁ = π₁∘A258746,     A258996∘π₂ = π₂∘A258746
                  # π₁∘A258996   =   A258746∘π₁, π₂∘A258996   =   A258746∘π₂
                  # A258746∘A258996 = A258996∘A258746
• HCS:            A002487∘π₁∘   A233279 / A002487∘(1+π₁∘A233279) = A071766 / A229742
                  A002487∘(1+π₂∘A233279) / A002487∘π₂∘   A233279 = A071766 / A229742
                  A002487∘(1+π₁∘A006068) / A002487∘π₁∘   A006068 = A071766 / A229742
                  A002487∘π₂∘   A006068 / A002487∘(1+π₂∘A006068) = A071766 / A229742
                  # For A233279 the orbit analysis is the same as that of A231551.
                  # For A006068 the orbit analysis is the same as that of A153154.
• Yurramendi-2:   A002487∘π₁∘   A180200 / A002487∘(1+π₁∘A180200) = A245325 / A245326
                  A002487∘(1+π₂∘A180200) / A002487∘π₂∘   A180200 = A245325 / A245326
                  A002487∘(1+π₁∘A154435) / A002487∘π₁∘   A154435 = A245325 / A245326
                  A002487∘π₂∘   A154435  / A002487∘(1+π₂∘A154435) = A245325 / A245326
                  # For A180200 the orbit analysis is the same as that of A231551.
                  # For A154435 the orbit analysis is the same as that of A153154.

Relationships between these permutations:

• <tbody> </tbody>

  |

  ∘

  |

  ϵ

  η

  τ2

  μ2

  π1

  π2

  A258746

  A165199

  A233279

  A006068

  A180200

  A154435

  |

  |

  ϵ

  |

  ϵ

  η

  τ2

  μ2

  π1

  π2

  A258746

  A165199

  A233279

  A006068

  A180200

  A154435

  |

  |

  η

  |

  η

  ϵ

  μ2

  τ2

  π2

  π1

  A165199

  A258746

  A006068

  A233279

  A154435

  A180200

  |

  |

  τ2

  τ2

  μ2

  ϵ

  η

  π1∘τ1

  π2∘τ1

  η∘A284120

  A284120

  τ2∘A233279

  τ2∘A006068

  τ2∘A180200

  τ2∘A154435

  |

  |

  μ2

  μ2

  τ2

  η

  ϵ

  μ2∘π1

  μ2∘π2

  A284120

  A165199∘μ2

  μ2∘A233279

  μ2∘A006068

  μ2∘A180200

  μ2∘A154435

  |

  |

  π1

  |

  π1

  π2

  τ1∘π1

  π1∘μ2

  ϵ

  η

  π2∘A165199

  π2∘A258746

  π2∘A006068

  π2∘A233279

  π2∘A154435

  π2∘A180200

  |

  |

  π2

  |

  π2

  π1

  τ1∘π2

  π2∘μ2

  η

  ϵ

  π1∘A165199

  π1∘A258746

  π1∘A006068

  π1∘A233279

  π1∘A154435

  π1∘A180200

  |

  |

  A258746

  |

  A258746

  A165199

  η∘A284120

  A284120

  A165199∘π2

  A165199∘π1

  ϵ

  η

  A180200

  A154435

  A233279

  A006068

  |

  |

  A165199

  |

  A165199

  A258746

  A284120

  μ2∘A165199

  A258746∘π2

  A258746∘π1

  η

  ϵ

  A154435

  A180200

  A006068

  A233279

  |

  |

  A233279

  |

  A233279

  A154435

  A006068

  A180200

  A154435∘π2

  A154435∘π1

  A154435∘A165199

  A154435∘A258746

  A154435∘A006068

  A154435∘A233279

  A154435∘A154435

  A154435∘A180200

  |

  |

  A006068

  |

  A006068

  A180200

  A233279

  A154435

  A180200∘π2

  A180200∘π1

  A180200∘A165199

  A180200∘A258746

  A180200∘A006068

  A180200∘A233279

  A180200∘A154435

  A180200∘A180200

  |

  |

  A180200

  |

  A180200

  A006068

  A154435

  A233279

  A006068∘π2

  A006068∘π1

  A006068∘A165199

  A006068∘A258746

  A006068∘A006068

  A006068∘A233279

  A006068∘A154435

  A006068∘A180200

  |

  |

  A154435

  |

  A154435

  A233279

  A180200

  A006068

  A233279∘π2

  A233279∘π1

  A233279∘A165199

  A233279∘A258746

  A233279∘A006068

  A233279∘A233279

  A233279∘A154435

  A233279∘A180200

  |

• A258746∘A258746 = A000027 (identity)
  A258746∘A180200 = A233279
  A258746∘A233279 = A180200
• A180200 = η∘A233279∘η, A233279 = η∘A180200∘η
  A154435 = η∘A006068∘η, A006068 = η∘A154435∘η
• A233279∘A258746 = A154435∘A165199 = A180200∘A284120
  A006068∘A258746 = A180200∘A165199 = A154435∘A284120
  A180200∘A258746 = A006068∘A165199 = A233279∘A284120
  A154435∘A258746 = A233279∘A165199 = A006068∘A284120

Inversely, A002487 can be obtained through all of these numerators and denominators and some permutations of ℕ, which are just the inverse of the permutations above. That is to say:

• A002487 = A007305∘ϵ∘π₁ = A162909∘A258746∘π₁ = A071766∘A233280∘π₁ = A245325∘A180201∘π₁.
• A002487 = A047679∘η∘π₁ = A162910∘A165199∘π₁ = A229742∘A003188∘π₁ = A245326∘A154436∘π₁.

 

Relationships between permutations of both classes:

• A258996 = π₁∘A258746∘π₁, A258746 = π₁∘A258996∘π₁,                    A332769 = π₁∘A165199∘π₁, A165199 = π₁∘A332769∘π₁
  A231551 = π₁∘A233279∘π₁, A233279 = π₁∘A231551∘π₁,                    A153154 = π₁∘A006068∘π₁, A006068 = π₁∘A153154∘π₁
  A284459 = π₁∘A180200∘π₁, A180200 = π₁∘A284459∘π₁,                    A154437 = π₁∘A154435∘π₁, A154435 = π₁∘A154437∘π₁

Between systems

Within classes

Structure of permutation system is the same in both classes (Class 1 / Class 2), ([54], [55]):

From                To	Calkin-Wilf/ Stern-Brocot	driB/Bird	Yu-Ting-1/HCS	Yurramendi-1/2
Calkin-Wilf/Stern-Brocot	ϵ	σ2	σ3	σ4
driB/Bird	σ2	ϵ	σ4	σ3
Yu-Ting-1/HCS	σ3'	σ4'	ϵ	σ2'
Yurramendi-1/2	σ4'	σ3'	σ2'	ϵ

ϵ ≡ A000027, the positive integers (identity permutation).
η ≡ A054429, the inverse permutation by blocks of 2^m terms.
τ₁ ≡ A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).
τ₂ ≡ A063946: write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.

• σ₂ ∘σ₂ = ϵ,            σ'₂∘σ'₂ = ϵ,           σ₂∘σ'₂ = σ'₂∘σ₂.
• σ₂ ∘η  = η ∘σ₂,          σ'₂∘η  = η ∘σ'₂.
• σ₂ ∘τ₁ = τ₁∘σ₂,         σ₂ ∘τ₂  = τ₂∘σ₂
  σ'₂∘τ₁ = τ₁∘σ'₂,          σ'₂∘τ₂ = τ₂∘σ'₂
• ({ϵ, η, τ₁, τ₂, σ₂, σ'₂, ...(*), η∘τ₁∘τ₂∘σ₂∘σ'₂}, ∘) is an elementary abelian group of order 2⁶.
• σ₃∘σ'₃ = ϵ,            σ'₃∘σ₃ = ϵ,          
  σ₄∘σ'₄ = ϵ,            σ'₄∘σ₄ = ϵ,          
  ∀m ≥ 0, ∀n > 0 such that n < 2^m+2 + 2^m+1, one has σ₃(n) = σ'₃^(2m-1)(n), and viceversa. It is the same for σ₄ and σ'₄.
• σ₃∘σ'₄ = σ₄∘σ'₃ = σ₂,     σ'₃∘σ₄  = σ'₄∘σ₃ = σ'₂
  σ₂∘σ₄  = σ₄∘σ'₂ = σ₃,      σ'₂∘σ'₄ = σ'₄∘σ₂ = σ'₃
  σ₂∘σ₃  = σ₃∘σ'₂ = σ₄,     σ'₂∘σ'₃ = σ'₃∘σ₂ = σ'₄
• (σ'₄∘σ'₃)∘(σ₄ ∘σ₃ ) = (σ'₃∘σ'₄)∘(σ₃ ∘σ₄ )
  (σ₄ ∘σ₃ )∘(σ'₄∘σ'₃) = (σ₃ ∘σ₄ )∘(σ'₃∘σ'₄)

Within Class 1

σ₂ ≡ A258996, σ'₂ ≡ A092569 ≡ η∘τ₁ = τ₁∘η,            σ₂ ≡ A332769, σ'₂ ≡ A092569 ≡ η∘τ₁ = τ₁∘η,
σ₃ ≡ A231551, σ'₃ ≡ A231550,                                σ₃ ≡ A153154, σ'₃ ≡ A153153,
σ₄ ≡ A284459, σ'₄ ≡ A284460,                                σ₄ ≡ A154437, σ'₄ ≡ A154438

• σ₂ ∘σ'₂ = σ'₂∘σ₂ = A284447.                                  σ₂ ∘σ'₂ = σ'₂∘σ₂
• σ₂ ∘η   =  η∘σ₂ = A332769, σ'₂∘η = η∘σ'₂ = τ₁.              σ₂ ∘η   =  η∘σ₂ = A258996, σ'₂∘η = η∘σ'₂ = τ₁.
• σ'₂∘τ₁  = τ₁∘σ'₂ = η
• σ₃∘σ₄  = A154437²         σ'₃∘σ'₄ = A153153²      ,          σ₃∘σ₄  = A284459²             σ'₃∘σ'₄ = A231550²
  σ₄∘σ₃  = A153154²         σ'₄∘σ'₃ = A154438²      ,          σ₄∘σ₃  = A231551²          σ'₄∘σ'₃ = A284460²
• σ₂∘σ'₃                                           =          σ₂∘σ'₄
  σ₂∘σ'₄                                           =          σ₂∘σ'₃  
• For σ'₃, σ₄ and σ'₄ the orbit analysis is the same as that of σ₃ (A231551).
• The enumeration system A002487∘ A092569 / A002487∘(1+A092569) is not within the scope of this work. These two enumeration systems A002487∘(1+A153153) / A002487∘A153153 and A002487∘A284460 / A002487∘(1+A284460) are the same, as well as these two systems A002487∘A231550 / A002487∘(1+A231550) and A002487∘(1+A154438) / A002487∘A154438, but they both are also not within the scope.

Permutations are the following ones:

• Calkin-Wilf and driB:      A002487∘σ₂  = A162911,  A162911∘σ₂  = A002487
                    A002487'∘σ₂ = A162912,  A162912∘σ₂  = A002487' (by coordinating carefully the n^th terms)
                    A007306∘σ₂  = A268087,  A268087∘σ₂  = A007306  (by coordinating carefully the n^th terms)
• Yu-Ting^-1 and Yurramendi-1:  A020651∘σ'₂ = A245327,  A245327∘σ'₂ = A020651
                    A020650∘σ'₂ = A245328,  A245328∘σ'₂ = A020650
                    A086592∘σ'₂ = A273493,  A273493∘σ'₂ = A086592
• Calkin-Wilf and Yu-Ting^-1:   A002487∘σ₃  = A020651,  A020651∘σ'₃ = A002487
                    A002487'∘σ₃ = A020650,  A020650∘σ'₃ = A002487' (by coordinating carefully the n^th terms)
                    A007306∘σ₃  = A086592,  A086592∘σ'₃ = A007306  (by coordinating carefully the n^th terms)
• driB and Yurramendi-1:     A162911∘σ₃  = A245327,  A245327∘σ'₃ = A162911
                    A162912∘σ₃  = A245328,  A245328∘σ'₃ = A162912
                    A268087∘σ₃  = A273493,  A273493∘σ'₃ = A268087
• Calkin-Wilf and Yurramendi-1: A002487∘σ₄  = A245327,  A245327∘σ'₄ = A002487
                    A002487'∘σ₄ = A245328,  A245328∘σ'₄ = A002487' (by coordinating carefully the n^th terms)
                    A007306∘σ₄  = A273493,  A273493∘σ'₄ = A007306  (by coordinating carefully the n^th terms)
• driB and Yu-Ting^-1:       A162911∘σ₄  = A020651,  A020651∘σ'₄ = A162911
                    A162912∘σ₄  = A020650,  A020650∘σ'₄ = A162912
                    A268087∘σ₄  = A086592,  A086592∘σ'₄ = A268087

Within Class 2

σ₂ ≡ A258746, σ'₂ ≡ A117120 ≡ η∘τ₂ = τ₂∘η,             σ₂ ≡ A165199, σ'₂ ≡ A117120 ≡ η∘τ₁ = τ₁∘η,
σ₃ ≡ A233279, σ'₃ ≡ A233280,                                  σ₃ ≡ A006068, σ'₃ ≡ A003188, 
σ₄ ≡ A180200, σ'₄ ≡ A180201,                                 σ₄ ≡ A154435, σ'₄ ≡ A154436

• σ₂ ∘σ'₂ = σ'₂∘σ₂ = A284120.                                   σ₂ ∘σ'₂ = σ'₂∘σ₂
• σ₂ ∘η   =  η∘σ₂ = A165199, σ'₂∘η = η∘σ'₂ = τ₂.               σ₂ ∘η   =  η∘σ₂ = A258746, σ'₂∘η = η∘σ'₂ = τ₂.
• σ'₂∘τ₂  = τ₂∘σ'₂ = η
• σ₃∘σ₄  = A154435²         σ'₃∘σ'₄ = A003188² = A064706,          σ₃∘σ₄  = A180200² = A180198   σ'₃∘σ'₄ = A233280²
  σ₄∘σ₃  = A006068² = A064707  σ'₄∘σ'₃ = A154436²       ,          σ₄∘σ₃  = A233279²          σ'₄∘σ'₃ = A180201² = A180199
• σ₂∘σ'₃                                            =          σ₂∘σ'₄
  σ₂∘σ'₄                                            =          σ₂∘σ'₃  
• For σ'₃, σ₄ and σ'₄ the orbit analysis is the same as that of σ₃ (A233279, that is to say A231551).
• The enumeration system A002487∘ A117120 / A002487∘(1+A117120) is not within the scope of this work. These two enumeration systems A002487∘(1+A003188) / A002487∘A003188 and A002487∘A180201 / A002487∘(1+A180201) are the same, as well as these two systems A002487∘A233280 / A002487∘(1+A233280) and A002487∘(1+A154436) / A002487∘A154436, but they both are also not within the scope.

Permutations are the following ones:

• Stern-Brocot and Bird:      A007305∘σ₂  = A162909,   A162909∘σ₂  = A007305 (by coordinating carefully the n^th terms)
                     A047679∘σ₂  = A162910,   A162910∘σ₂  = A047679
                     A007306∘σ₂  = A268087,    A268087∘σ₂  = A007306 (by coordinating carefully the n^th terms)
• HCS and Yurramendi-2:       A071766∘σ'₂ = A245325,   A245325∘σ'₂ = A071766
                     A229742∘σ'₂ = A245326,   A245326∘σ'₂ = A229742
                     A071585∘σ'₂ = A273494,   A273494∘σ'₂ = A071585
• Stern-Brocot and HCS:       A007305∘σ₃ = A071766,   A071766∘σ'₃ = A007305 (by coordinating carefully the n^th terms)
                     A047679∘σ₃ = A229742,   A229742∘σ'₃ = A047679
                     A007306∘σ₃  = A071585,    A071585∘σ'₃ = A007306 (by coordinating carefully the n^th terms)
• Bird and Yurramendi-2:      A162909∘σ₃ = A245325,   A245325∘σ'₃ = A162909
                     A162910∘σ₃ = A245326,   A245326∘σ'₃ = A162910
                     A268087∘σ₃  = A273494,    A273494∘σ'₃ = A268087
• Stern-Brocot and Yurramendi-2: A007305∘σ₄ = A245325,   A245325∘σ'₄ = A007305 (by coordinating carefully the n^th terms)
                     A047679∘σ₄ = A245326,   A245326∘σ'₄ = A047679
                     A007306∘σ₄  = A273494,    A273494∘σ'₄ = A007306 (by coordinating carefully the n^th terms)
• Bird and HCS:           A162909∘σ₄  = A071766,   A071766∘σ'₄ = A162909
                     A162910∘σ₄  = A229742,   A229742∘σ'₄ = A162910
                     A268087∘σ₄  = A071585,    A071585∘σ'₄ = A268087

• σ_2-1∘ σ_2-2 = σ_2-2∘σ_2-1,        σ'_2-1∘σ'_2-2 = σ'_2-2∘σ'_2-1

Between classes

Between Class 1 and Class 2

π₁ ≡ A059893, the bit-reversal permutation by blocks of 2^m terms.
π₂ ≡ A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.

• Calkin-Wilf and Stern-Brocot: A007305∘π₁ = A002487,  A002487∘π₁ = A007305 (by coordinating carefully the n^th term)
                                A047679∘π₁ = A002487', A002487'∘π₁= A047679
                                A007306∘π₁ = A007306                        (by coordinating carefully the n^th term)
                                A007305∘π₂ = A002487', A002487'∘π₂= A007305 (by coordinating carefully the n^th term)
                                A047679∘π₂ = A002487,  A002487∘π₂ = A047679
                                A007306∘π₂ = A007306                        (by coordinating carefully the n^th term)
• driB and Bird:                A162909∘π₁ = A162911,  A162911∘π₁ = A162909
                                A162910∘π₁ = A162912,  A162912∘π₁ = A162910
                                A268087∘π₁ = A268087
                                A162909∘π₂ = A162912,  A162912∘π₂ = A162909
                                A162910∘π₂ = A162911,  A162911∘π₂ = A162910
                                A268087∘π₂ = A268087
• Yu-Ting and HCS:              A071766∘π₁ = A020651,  A020651∘π₁ = A071766
                                A229742∘π₁ = A020650,  A020650∘π₁ = A229742
                                A071585∘π₁ = A086592,  A086592∘π₁ = A071585
                                A071766∘π₂ = A245328,  A245328∘π₂ = A071766
                                A229742∘π₂ = A245327,  A245327∘π₂ = A229742
                                A071585∘π₂ = A273493,  A273493∘π₂ = A071585
• Yurramendi-1 and -2:          A245325∘π₁ = A245327,  A245327∘π₁ = A245325
                                A245326∘π₁ = A245328,  A245328∘π₁ = A245326
                                A273494∘π₁ = A273493,  A273493∘π₁ = A273494
                                A245325∘π₂ = A020650,  A020650∘π₂ = A245325
                                A245326∘π₂ = A020650,  A020650∘π₂ = A245326
                                A273494∘π₂ = A086592,  A086592∘π₂ = A273494

Frequencies on A002487 and A007306

Because the numerator and denominator of the systems are permutations of A002487, if the order in each block is not taken into account, then an issue raises about the frequencies of each integer in each block; this issue concerns all of them.

Let s(n,m) be the function that gives the frequency of integer n > 1 at each block m ≥ 0 of sequence A002487 ([56]).

• ∀n > 2, ∀m ≥ 0, s(n,m) %% 2 = 0. ∀m ≥ 0, s(1, m) = f(2, m) = 1.
• ∀m ≥ 0, ∑_{n > 1} s(n, m) = 2^m (A000079).
• ∀n ≥ 1, ∀m ≥ n - 1, s(n, m) = s(n, n - 1).
• ∀n ≥ 1, s(n, n - 1) = φ(n), where φ is the Euler's totient function, A000010 ([57], fact no.12).
• ∀m ≥ 0, max{n | s(n, m) > 0} = F(m + 2), where F is the Fibonacci sequence, A000045 ([58], Theorem 2.1). Morever, ∀m ≥ 2 s(F(m + 2), m) = 2.
• It seems that ∀m ≥ 6 the number of gaps (s(m, n) = 0) before s(F(m + 2), m) are related to Fibonacci sequence.

In the same way, sequences num+den are permutations of A007306 (it is A002487 + A002487' (Calkin-Wilf) and A007305 + A047679 (Stern-Brocot)).
Let f(n,m) be the function that gives the frequency of integer n > 1 at each block m ≥ 0 in sequence A007306 ([59]).

• ∀n > 1, ∀m ≥ 0, f(n,m) %% 2, except f(2, 0) = 1.
• ∀m ≥ 0, ∑_{n > 1} f(n, m) = 2^m (A000079).
• ∀m ≥ 0, ∑_{n > 1} n·f(n, m) = 2·3^m (2·A000244).
• ∀n > 1, ∑_{m ≥ 0} f(n, m) = φ(n)
• ∀m ≥ 0, min{n | f(n, m) > 0} = m + 2. ([60], fact no.11).
• ∀m ≥ 0, max{n | f(n, m) > 0} = F(m + 1). Morever, f(F(m + 1), m) = 2.
• ∀n > 1, ∀m ≥ 0, f(n, m) = s(n, m + 1) - s(n, m).
• ∀n > 1, f(n, n - 2) = 2. It is a straight line.
  ∀n > 1, max{m | f(n, m) > 0} = n - 2.
  More linear relationships can be observed ([61]; [62]).
• A procedure can be defined in order to sweep the table by searching for straight lines. A start point of a straight line is the pair (n, m) with minimum m. The first start point is (n, m) = (2, 0), then (3,1), (4,2), (5,2), (5,3), (7,3), (8,3), ..., from bottom to up and left to right. The effect of a line is just to subtract it 'characteristic value' from the values of points in the straight line. When a new value results 0, the corresponding point is 'disappeared'. ([63]).
• Start points can be represented in the plan (n,m). Some quadratic relations can be observed ([64], [65]).