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# User:Yosu Yurramendi

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Yosu Yurramendi Mendizabal.
Donostia ([1]), Basque Country ([2]), 1955.

BSc in Mathematics, Universidad Complutense de Madrid, 1977.
Ph.D. in Statistics, Université Pierre et Marie Curie, Paris VI, 1984 ([3]).

University of the Basque Country ([4]).
Department of Computing Sciences and Artificial Intelligence ([5]).
Main field of study: Data Analysis.
yosu.yurramendi at ehu.eus

Main contributions to OEIS:

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

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]).

A002487 can be represented by blocks of 2m terms (m ≥ 0) in a natural way:   1  1 2  1 3 2 3  1 4 3 5 2 5 3 4 ... ([7], [8]). This representation is basic for this work.

• The classes of the well-known enumeration systems has been generated 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}: [9],  [10].
• More classes could be defined [11]).
• Graphic representation of ℚ+, and cover by blocks based on Stern's sequence: [12]. An enumeration system is represented by a trajectory on the cover.
• More generally, let σ be an infinite permutation (bijection) on ℕ such that it is itself a composition of finite-range permutations,
where everyone works within a closed interval of {2m, ..., 2m+1-1} (m ≥ 0), which is composed by 2m terms.
• ∀m ≥ 0, ∀n > 0 such that 2m ≤ n < 2m+1 σ(2m)(n) = n.
• ∀m ≥ 0, ∀n > 0 such that    n < 2m+1 σ(2m)(n) = n.
• If σ(2) = 2 and σ(3) = 3 (alternatively, σ(2) = 3 and σ(3) = 2), then A002487σ / A002487∘(1+σ) (alternatively, A002487∘(1+σ) / A002487σ) constitutes an enumeration system for positive rationals. (draft of the proof [13])
• In this sense each considered well-known enumeration could be characterized by A002487 and a particular permutation σ. See below which they are.
• The fast algorithm for A002487 beside a fast algorithm for the particular permutation σ of an enumeration system give the irreducible fraction corresponding to a given location.
• In the enumeration systems considered every numerator (denominator) is a permuted sequence of A002487,
where the permutation is produced at each block of 2m terms (see below).
• Sequences num+den (= numerator+denominator) are considered.
• numerator/num+den and denominator/num+den are enumeration systems of rationals in (0,1).
• Sequence built by sums of blocks of every num+den sequence is 2·3m (2·A000244).
• numerator and denominator can be computed from num+den (see below).
• Let f(n,m) be the function that gives the frequency of integer n > 1 at each block m ≥ 0 in sequences num+den ([14]).
• ∀n > 1, ∑m ≥ 0 f(n,m) = φ(n) (φ, Euler's totient function, A000010). ([15], fact no.12). That is to say, each integer n > 1 appears φ(n) times in sequences num+den.
• ∀n > 1, max{m ≥ 0 | f(n,m) > 0} = n-2 ([16], fact no.11).
• ∀m ≥ 0, ∑n > 1 n·f(n,m) = 2·3m.
• ∀m ≥ 0, min{n > 1 | f(n,m) > 0} = m+2.
• ∀m ≥ 0, max{n > 1 | f(n,m) > 0} = F(m+3), where F is the Fibonacci sequence, A000045 ([17], Theorem 2.1). Morever, f(F(m+3), m) = 2.
• Interesting linear relationships can be observed ([18]).
• 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) the nature of the underlying bijection:
• What location corresponds to a given irreducible fraction (see below).
• And, viceversa, what irreducible fraction corresponds to a given location.
• Same questions can be considered for numerator, denominator, and num+den sequences.
• In order to compute the value of its nth term, beyond the recursive formula of its definition, a fast algorithm for A002487 has been defined (it is the base for a well-known enumeration system, both for numerator and for denominator): given n > 0, it computes A002487(n) by taking into account the binary representation of n ([19]).
A very similar fast algorithm for A007306 has also been defined (it is the base for the num+den of two well-known enumeration systems): [20].

### Classification

#### Class 1

numerator(n)                         num+den(n)
∀n > 0    f(n) = --------------             f(n)+1 = --------------
denominator(n)                      denominator(n)

• Calkin-Wilf ([21]):     f(2n  ) =  f(n)    / (f(n)+1) = numerator(n)   / num+den(n)
f(2n+1) = (f(n)+1) /       1  = num+den(n)     / denominator(n)
• driB ([22]):         f(2n  ) =       1  / (f(n)+1) = denominator(n) / num+den(n)
f(2n+1) = (f(n)+1) /  f(n)    = num+den(n)     / numerator(n)
• Yu-Ting-1 ([23], [24]):  f(2n  ) =  f(n)    / (f(n)+1) = numerator(n)   / num+den(n)
f(2n+1) = (f(n)+1) /  f(n)    = num+den(n)     / numerator(n)
• Yurramendi-1:          f(2n  ) =       1  / (f(n)+1) = denominator(n) / num+den(n)
f(2n+1) = (f(n)+1) /  f(n)    = num+den(n)     / denominator(n)

Generation of the four enumeration systems (R code, [25]): [26].

Graphic representation of enumeration systems in the plane of ℚ+: [27] .

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              Yurramendi-1 |
f(2n+1) =                 |                                        |
(f(n)+1)/f(n)  |     Yu-Ting-1                  driB    |
------------------------------------------

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

Sequences in OEIS:

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 [28] for involved relations):

• Calkin-Wilf  : numerator(2q*(2n+1)    ) = denominator(2q*(2n+1) - 1) = num+den(n), n > 0, q 0.

= A002487A153152,  A153151A153152 = A153152A153151 = A000027.  (implicit relationship).
numerator(2q           ) = denominator(2q*2      - 1) = 1,            q 0.
A002487A000079          = A000225           = 1
• driB         : numerator(2q*(2n+1) - (-2)q*2/3                   - 1/3) = denominator(2q*(2n+1) + (-2)q*2/3                   - 2/3) = num+den(n), n > 0, q 0.

A334999 = A334999A334998 = A000027 .
numerator(2q        - (-2)q*1/3 - [2q-2 - (-2)q-2] - 2/3) = denominator(2q        + (-2)q*1/3 - [2q-2 + (-2)q-2] - 1/3)        = 1,q 0.

A162911A061547                                          = A162912A086893 .                                          = 1
• Yu-Ting      : numerator(2q*(2n+1)    ) = denominator(    2n       ) = num+den(n), n > 0, q = 0.
numerator(
2q*(2n+1)    ) = denominator(2q*(2n+1) + 1) = num+den(n), n > 0, q > 0.

A020650A065190 ,    A020650 = A020651A065190, A065190A065190 = A000027 .
numerator(2q           ) = denominator(2q         + 1) = 1,                 q
> 0. numerator(1) = denominator(1) = 1.

A020651A000079          = A083318            = 1
• Yurramendi-1 : numerator(   (2n+1)    ) = denominator(2q*(2n+1) - 1) = num+den(n), n > 0, q = 0.
numerator(2q*(2n+1) - 2) = denominator(2q*(2n+1) - 1) = num+den(n), n > 0, q > 0.

A245328 = A245327A065190, A065190A065190 = A000027.
numerator(
2q*2      - 2) = denominator(2q*2      - 1) = 1,             q > 0. numerator(1) = denominator(1) = 1.

A245327A095121          = A000225            = 1

##### Given a fraction, compute its location in a system

It can be computed by a procedure based on an Euclidean algorithm ([29]), by copying and pasting the following code (R, [30]): [31]

#### Class 2

∀m ≥ 0, ∀k such that 0 ≤ k < 2m

numerator(2m+k)                          num+den(2m+k)
f(2m+k) = -----------------           f(2m+k)+1 = -----------------
denominator(2m+k)                       denominator(2m+k)

• Stern-Brocot ([32]): f(2m+1   +k) =  f(2m+k)    / (f(2m+k)+1) = numerator(2m+k)   / num+den(2m+k)
f(2m+1+2m+k) = (f(2m+k)+1) /          1  = num+den(2m+k)     / denominator(2m+k)
• Bird ([33]):         f(2m+1   +k) =          1  / (f(2m+k)+1) = denominator(2m+k) / num+den(2m+k)
f(2m+1+2m+k) = (f(2m+k)+1) /  f(2m+k)    = num+den(2m+k)     / numerator(2m+k)
• HCS ([34]):          f(2m+1   +k) =  f(2m+k)    / (f(2m+k)+1) = numerator(2m+k)   / num+den(2m+k)
f(2m+1+2m+k) = (f(2m+k)+1) /  f(2m+k)    = num+den(2m+k)     / numerator(2m+k)
• Yurramendi-2:          f(2m+1   +k) =          1  / (f(2m+k)+1) = denominator(2m+k) / num+den(2m+k)
f(2m+1+2m+k) = (f(2m+k)+1) /  f(2m+k)    = num+den(2m+k)     / denominator(2m+k)

Generation of the four enumeration systems (R code, [35]): [36].

Graphic representation of enumeration systems in the plane of ℚ+: [37] .

The four systems organized in a table

f(1) = 1/1. ∀m ≥ 0, ∀k such that 0 ≤ k < 2m

f(2m+1+k) =    f(2m+k)/(f(2m+k)+ 1)      1/(f(2m+k)+1)
-----------------------------------------
(f(2m+k)+1)/1       |  Stern-Brocot           Yurramendi-2  |
f(2m+1+2m+k) =                        |                                       |
(f(2m+k)+1)/f(2m+k)  |      HCS                     Bird     |
-----------------------------------------

For all four systems:   ∀m ≥ 0, ∀k 0 ≤ k < 2m, f(2m+1 + k) < 1, f(2m+1 + 2m + k) > 1.

The numerator and denominator of four systems organized in a table

numerator(1) = 1, denominator(1) = 1. ∀m ≥ 0, ∀k such that 0 ≤ k < 2m

numerator(2m+1+k) =         numerator(2m+k)    denominator(2m+k)
---------------------------------------
denominator(2m+k) |    Stern-Brocot       Yurramendi-2  |
denominator(2m+1+2m+k) =                    |                                     |
numerator(2m+k) |       HCS                Bird       |
---------------------------------------

numerator(2m+1+2m+k) = denominator(2m+1+k) = num+den(2m+k).

Sequences in OEIS:

The relationships between numerator, denominator, and num+den are more general than the previous equalities numerator(2m+1+2m+k) = denominator(2m+1+k) = num+den(2m+k).(it is the case q = 1) (see R code [38] for involved relations):

• Stern-Brocot : numerator((  2q + 1)*2m + k) = denominator((  2q - 1)*2*2m + k) = num+den(2m + k), m ≥ 0, 0 ≤ k < 2m, q > 0.

A047679 = A007305A153142A153141A153142 = A153142A153141 = A000027.   (implicit relationship).
numerator(   2q-1          ) = denominator(   2q-1    *2    - 1) = 1,                                 q > 0.
A007305A000079              = A047679A000225                  = 1
• Bird         : numerator((8*2q - 3 - 5*(-1)q)*1/6*2m + k) = denominator((5*2q - 3 - 5*(-1)q)*1/6*2m + k) = num+den(2m + k), m ≥ 0, 0 ≤ k < 2m, q > 0.

,                  A162910 = A162909A154447A154448A154447 = A154447A154448 = A000027.
numerator((4*2q - 3 -   (-1)q)*1/6)        = denominator( 5*2q - 3 +   (-1)q)*1/6)        = 1, q > 0.

A162909A000975                            = A162910A081254 .                              = 1
• HCS          : numerator((  2q + 1)*2m + k) = denominator(           2*2m + k) = num+den(2m + k), m ≥ 0, 0 ≤ k < 2m, q = 1.
numerator((  2q + 1)*2m + k) = denominator((3*2q + 1)*  2m + k) = num+den(2m + k), m ≥ 0, 0 ≤ k < 2m, q > 1.

A229742 = A071766A063946, A063946A063946 = A000027.
numerator(   2q            ) = denominator( 3*2q-1            ) = 1,                                  q > 0. numerator(1) = denominator(1) = 1.

A071766A000079              = A003945                 = 1
• Yurramendi-2 : numerator(         3*2m + k) = denominator((  2q - 1)*2*2m + k) = num+den(2m + k), m ≥ 0, 0 ≤ k < 2m, q = 1.
numerator((3*2q - 2)*2m + k) = denominator((  2q - 1)*2*2m + k) = num+den(2m + k), m ≥ 0, 0 ≤ k < 2m, q > 1.

A245326 = A245325A063946, A063946A063946 = A000027.
numerator( 3*2q-1 - 1      ) = denominator(   2q     *2    - 1) = 1,                                  q > 0. numerator(1) = denominator(1) = 1.

A245325A083329              = A000225                 = 1

##### Given a fraction, compute its location in a system

It can be computed by a procedure based on an Euclidean algorithm ([39]), by copying and pasting the following code (R, [40]): [41]

### Permutations

They all can be computed by copying and pasting the following code (R, [42]): [43].

Some useful general permutations of the positive integers:
ϵ ≡ A000027, the positive integers (identity permutation).
η ≡ A054429, the inverse permutation by blocks of 2m terms.
τ1A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1), n > 0.
τ2A063946: 1 is fixed, followed by infinite number of adjacent transpositions (2m+1+k, 2m+1+2m+k), m ≥ 0, 0 ≤ k < 2m.

• ηη = ϵ.
({ϵ, η}, ∘) is a group (C2, [44]).
• τ1τ1 = τ2τ2 = ϵ.
ητ1 = τ1η = A092569, ητ2 = τ2η = A117120.
({ϵ, η, τ1, ητ1}, ∘) is a Klein 4-group (C2xC2, [45]), as well as ({ϵ, η, τ2, ητ2}, ∘).
• τ1τ2 = τ2τ1.
({ϵ, η, τ1, τ2, ητ1, ητ2, τ1τ2, ητ1τ2}, ∘) is an elementary abelian group of order 23 (C2xC2xC2, [46]).
• For η, τ1, τ2, A092569, A117120 ∀m > 1, there are 2m-1 orbits of length 2. For m = 0 one fixed point, and for m = 1 two fixed points. Therefore, they all are self-inverse.

π1A059893, the bit-reversal permutation by blocks of 2m terms.
π2A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.

• π1π1 = π2π2 = ϵ.
π1π2 = π2π1 = η, π1η = ηπ1 = π2, π2η = ηπ2 = π1.
({ϵ, η, π1, π2}, ∘) is a Klein 4-group (C2xC2, [47]).
• 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.
• π1τ1 = τ2π1 , τ1π1 = π1τ2
π2τ1 = τ2π2 , τ1π2 = π2τ2

#### Numerator and denominator of systems are permuted sequences of A002487

##### Class 1

ϵ ≡ A000027, the positive integers (identity permutation).
η ≡ A054429, the inverse permutation by blocks of 2m terms.
π1A059893, the bit-reversal permutation by blocks of 2m terms.
π2A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.
τ1A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).

• Calkin-Wilf:   A002487ϵ         / A002487∘(1+ϵ)     = A002487 / A002487'
A002487∘(1+ηϵ)     / A002487ηϵ      = A002487 / A002487'
• driB:           A002487∘   A258996 / A002487∘(1+A258996) = A162911 / A162912
A002487∘(1+ηA258996) / A002487η∘ A258996  = A162911 / A162912
A002487∘(1+ A332769) / A002487∘  A332769  = A162911 / A162912
# ({ϵ, η, τ1, ητ1,   A258996, η∘ A258996,  τ1A258996, ητ1A258996}, ∘)
# ({ϵ, η, τ1, ητ1, η∘  A332769,  A332769, ητ1A332769,  τ1A332769}, ∘)
# ({ϵ, η, τ1, ητ1, ητ1A284447, τ1A284447, η∘  A284447,    A284447}, ∘)
# they all three are the same elementary abelian group of order 23 (C2xC2xC2, [48]).
# A002487A284447 / A002487∘(1+A284447) is not 'driB' system.
# ητ1 = τ1η = A258996A284447 = A284447A258996 = A092569,
# For A258996, A332769, A284447, ∀m > 1, there are 2m-1 orbits of length 2. Therefore, they all three are self-inverse.
# ∀m ≥ 0, ∀k 0 ≤ k < 2m, if m odd  π1A258996(2m+k) = A258996π1(2m+k),
if m even π1A258996(2m+k) = A258996π1(2m+1-1-k)
# ∀m ≥ 0, ∀k 0 ≤ k < 2m, if m odd  π2A258996(2m+k) = A258996π2(2m+k),
if m even π2A258996(2m+k) = A258996π2(2m+1-1-k)
• Yu-Ting:        A002487∘   A231551 / A002487∘(1+A231551) = A020651 / A020650
A002487∘(1+ηA231551) / A002487η∘ A231551  = A020651 / A020650
A002487∘(1+ A153154) / A002487∘  A153154  = A020651 / A020650

# For A231551
∀m > 0, there are 2 fixed points in positions 2m and 2m+2m-1,
∀m > 1, there is an unique orbit of length 2, in positions 2m+2m-2 and 2m+1-2m-2,
∀m > 2 (m = 2q + r, 1 ≤ r ≤ 2q) and ∀p > 0 such that 2p 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)
= 2A000295(p-1)*(2r   -1) if p = (q+1)
= 2A000295(p-1)*(22(p-1)-1) if p < (q+1)
∀m ≥ 0 there are altogether A007886(m) orbits.
∀m ≥ 0, ∀n > 0 such that n ≤ 2m+2 + 2m+1 (higher than 2m), one has A231551(2m)(n) = n.

# For A153154
∀m > 1 (m = 2q + r, 0 ≤ r < 2q), there are 2(A000295(q)+r) orbits of length 2q+1.
∀m > 0 there are A054243(m) orbits.
∀m ≥ 0, ∀n > 0 such that n ≤ 2m, one has A153154(2m)(n) = n.

• Yurramendi-1:   A002487∘   A284459 / A002487∘(1+A284459) = A245327 / A245328
A002487∘(1+ηA284459) / A002487η∘ 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 π1 π2 τ1 A258996 A332769 A231551 A153154 A284459 A154437 | | ϵ | ϵ η π1 π2 τ1 A258996 A332769 A231551 A153154 A284459 A154437 | | η | η ϵ π2 π1 A092569 A332769 A258996 A153154 A231551 A154437 A284459 | | π1 | π1 π2 ϵ η π1∘τ1=τ2∘π1 π2∘A332769 π2∘A258996 π2∘A153154 π2∘A231551 π2∘A154437 π2∘A284459 | | π2 | π2 π1 η ϵ π2∘τ1=τ2∘π2 π1∘A332769 π1∘A258996 π1∘A153154 π1∘A231551 π1∘A154437 π1∘A284459 | | τ1 | τ1 A092569 τ1∘π1=π1∘τ2 τ1∘π2=π2∘τ2 ϵ η∘A284447 A284447 τ1∘A231551 τ1∘A153154 τ1∘A284459 τ1∘A154437 | | A258996 | A258996 A332769 A332769∘π2 A332769∘π1 η∘A284447 ϵ η A284459 A154437 A231551 A153154 | | A332769 | A332769 A258996 A258996∘π2 A258996∘π1 A284447 η ϵ A154437 A284459 A153154 A231551 | | A231551 | A231551 A154437 A154437∘π2 A154437∘π1 A153154 | | A153154 | A153154 A284459 A284459∘π2 A284459∘π1 A231551 | | A284459 | A284459 A153154 A153154∘π2 A153154∘π1 A154437 | | A154437 | A154437 A231551 A231551∘π2 A231551∘π1 A284459 |

##### Class 2

ϵ ≡ A000027, the positive integers (identity permutation).
η ≡ A054429, the inverse permutation by blocks of 2m terms.
π1A059893, the bit-reversal permutation by blocks of 2m terms.
π2A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.
τ2A063946: write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.

Relationships between these permutations:

•  | ∘ | ϵ η π1 π2 τ2 A258746 A165199 A233279 A006068 A180200 A154435 | | ϵ | ϵ η π1 π2 τ2 A258746 A165199 A233279 A006068 A180200 A154435 | | η | η ϵ π2 π1 A117120 A258746 A165199 A006068 A233279 A154435 A180200 | | π1 | π1 π2 ϵ η π1∘τ2=τ1∘π1 π2∘A165199 π2∘A258746 π2∘A006068 π2∘A233279 π2∘A154435 π2∘A180200 | | π2 | π2 π1 η ϵ π2∘τ2=τ1∘π2 π1∘A165199 π1∘A258746 π1∘A006068 π1∘A233279 π1∘A154435 π1∘A180200 | | τ2 | τ2 A117120 τ2∘π1=π1∘τ1 τ2∘π2=π2∘τ1 ϵ η∘A284120 A284120 τ2∘A233279 τ2∘A006068 τ2∘A180200 τ2∘A154435 | | A258746 | A258746 A165199 A165199∘π2 A165199∘π1 η∘A284120 ϵ η A180200 A154435 A233279 A006068 | | A165199 | A165199 A258746 A258746∘π2 A258746∘π1 A284120 η ϵ A154435 A180200 A006068 A233279 | | A233279 | A233279 A154435 A154435∘π2 A154435∘π1 A006068 | | A006068 | A006068 A180200 A180200∘π2 A180200∘π1 A233279 | | A180200 | A180200 A006068 A006068∘π2 A006068∘π1 A154435 | | A154435 | A154435 A233279 A233279∘π2 A233279∘π1 A180200 |

#### Permutations between numerator and denominator of systems

##### Class 1

η ≡ A054429, the inverse permutation by blocks of 2m terms.
τ1A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).

##### Class 2

η ≡ A054429, the inverse permutation by blocks of 2m terms.
τ2A063946: write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.

#### Permutations between systems

##### Within classes

Structure of permutation system is the same in both classes (Class 1 / Class 2), ([50], [51]):

 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 2m terms.
τ1A065190: 1 is fixed, followed by infinite number of adjacent transpositions (n n+1).
τ2A063946: write n in binary and complement second bit (from the left), with a(0)=0 and a(1)=1.

• σ2 σ2 = ϵ,            σ'2σ'2 = ϵ,           σ2σ'2 = σ'2σ2.
• σ2η  = η σ2,          σ'2η  = ησ'2.
• σ2τ1 = τ1σ2,         σ2 τ2  = τ2σ2
σ'2τ1 = τ1σ'2,          σ'2τ2 = τ2σ'2
• ({ϵ, η, τ1, τ2, σ2, σ'2, ...(*), ητ1τ2σ2σ'2}, ∘) is an elementary abelian group of order 26.
• σ3σ'3 = ϵ,            σ'3σ3 = ϵ,
σ4σ'4 = ϵ,            σ'4σ4 = ϵ,
∀m ≥ 0, ∀n > 0 such that n < 2m+2 + 2m+1, one has σ3(n) = σ'3(2m-1)(n), and viceversa. It is the same for σ4 and σ'4.
• σ3σ'4 = σ4σ'3 = σ2,     σ'3σ4  = σ'4σ3 = σ'2
σ2σ4  = σ4σ'2 = σ3,      σ'2σ'4 = σ'4σ2 = σ'3
σ2σ3  = σ3σ'2 = σ4,     σ'2σ'3 = σ'3σ2 = σ'4
• (σ'4σ'3)∘(σ4σ3 ) = (σ'3σ'4)∘(σ3σ4 )
(σ4σ3 )(σ'4σ'3) = (σ3σ4 )
(σ'3σ'4)
###### Within Class 1

σ2A258996, σ'2A092569ητ1 = τ1η,            σ2A332769, σ'2A092569ητ1 = τ1η,
σ3A231551, σ'3A231550,                                σ3A153154, σ'3A153153,
σ4A284459, σ'4A284460,                                σ4A154437, σ'4A154438

• σ2 σ'2 = σ'2σ2 = A284447.                                  σ2 σ'2 = σ'2σ2
• σ2η   =  ησ2 = A332769, σ'2η = ησ'2 = τ1.              σ2η   =  ησ2 = A258996, σ'2η = ησ'2 = τ1.
• σ'2τ1  = τ1σ'2 = η
• σ3σ4  = A1544372         σ'3σ'4 = A1531532      ,          σ3σ4  = A2844592            σ'3σ'4 = A2315502
σ4σ3  = A1531542         σ'4σ'3 = A1544382      ,          σ4σ3  = A2315512          σ'4σ'3 = A2844602
• σ2σ'3                                           =          σ2σ'4
σ2σ'4                                           =          σ2σ'3
• For σ'3, σ4 and σ'4 the orbit analysis is the same as that of σ3 (A231551).
• The enumeration system A002487A092569 / A002487∘(1+A092569) is not within the scope of this work. These two enumeration systems A002487∘(1+A153153) / A002487A153153 and A002487A284460 / A002487∘(1+A284460) are the same, as well as these two systems A002487A231550 / A002487∘(1+A231550) and A002487∘(1+A154438) / A002487A154438, but they both are also not within the scope.

Permutations are the following ones:

###### Within Class 2

σ2A258746, σ'2A117120ητ2 = τ2η,             σ2A165199, σ'2A117120ητ1 = τ1η,
σ3A233279, σ'3A233280,                                  σ3A006068, σ'3A003188,
σ4A180200, σ'4A180201,                                 σ4A154435, σ'4A154436

• σ2 σ'2 = σ'2σ2 = A284120.                                   σ2 σ'2 = σ'2σ2
• σ2η   =  ησ2 = A165199, σ'2η = ησ'2 = τ2.               σ2η   =  ησ2 = A258746, σ'2η = ησ'2 = τ2.
• σ'2τ2  = τ2σ'2 = η
• σ3σ4  = A1544352         σ'3σ'4 = A0031882 ,          σ3σ4  = A1802002   σ'3σ'4 = A2332802
σ4σ3  = A0060682   σ'4σ'3 = A1544362       ,          σ4σ3  = A2332792          σ'4σ'3 = A1802012
• σ2σ'3                                            =          σ2σ'4
σ2σ'4                                            =          σ2σ'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 A002487A117120 / A002487∘(1+A117120) is not within the scope of this work. These two enumeration systems A002487∘(1+A003188) / A002487A003188 and A002487A180201 / A002487∘(1+A180201) are the same, as well as these two systems A002487A233280 / A002487∘(1+A233280) and A002487∘(1+A154436) / A002487A154436, but they both are also not within the scope.

Permutations are the following ones:

• σ2-1∘ σ2-2 = σ2-2σ2-1,        σ'2-1σ'2-2 = σ'2-2σ'2-1

##### Between classes
###### Between Class 1 and Class 2

π1A059893, the bit-reversal permutation by blocks of 2m terms.
π2A059894, Complement and reverse the order of all but the most significant bit in binary expansion of n.

## Number of binary pattern classes in the (m,n)-rectangular grid with k '1's and (mn-k) '0's

Two binary patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation .
A034851 : (1,n,k) triangle is the Losanitsch's triangle .
A226048 : (2,n,k) triangle .
A226290 : (3,n,k) triangle .
A225812 : (4,n,k) triangle (with María Merino).
A228022 : (5,n,k) triangle (with María Merino).
A228165 : (6,n,k) triangle (with María Merino).
A228166 : (7,n,k) triangle (with María Merino).
A228167 : (8,n,k) triangle (with María Merino).
A228168 : (9,n,k) triangle (with María Merino).
A228169 : (10,n,k) triangle (with María Merino).

A225826 to A225834 : (m,n) sequences, 1 < m < 11 (one by one).
A225910 : (m,n) table, 1 < m < 11 ((m,n) sequences all together).

YURRAMENDI MENDIZABAL Y. 2013. "Matematika esperimentalaren adibide bat: Lauki sareko patroi bitarren kopuruaren kalkulua", EKAIA, 26, 325-348] ([52]).
MERINO MAESTRE M., YURRAMENDI MENDIZABAL Y. 2014. "Lauki sareko patroi bitarren kalkulua, oinarrizko konbinatoriaren eskutik" EKAIA, 27, 237-262 ([53]).
MERINO MAESTRE M., UNANUE GUAL I. 2018. "Lauki sareko patroien kalkulua, Polyaren teoriaren eskutik" EKAIA, 34, 289-316 ([54]).