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# User:Yosu Yurramendi

Yosu Yurramendi Mendizabal.
Donostia (), Basque Country (), 1955.

Ph.D. in Statistics, Université Pierre et Marie Curie, Paris VI, 1984 ().

University of the Basque Country ().
Department of Computing Sciences and Artificial Intelligence ().
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' ().

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 ... (, ). 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}: ,  .
• More classes could be defined ).
• Graphic representation of ℚ+, and cover by blocks based on Stern's sequence: . 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 )
• 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 ().
• ∀n > 1, ∑m ≥ 0 f(n,m) = φ(n) (φ, Euler's totient function, A000010). (, 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 (, 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 (, Theorem 2.1). Morever, f(F(m+3), m) = 2.
• Interesting linear relationships can be observed ().
• 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 ().
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): .

### Classification

#### Class 1

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

• Calkin-Wilf ():     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 ():         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 (, ):  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, ): .

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

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  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 (), by copying and pasting the following code (R, ): 

#### 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 (): 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 ():         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 ():          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, ): .

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

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  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 (), by copying and pasting the following code (R, ): 

### Permutations

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

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, ).
• τ1τ1 = τ2τ2 = ϵ.
ητ1 = τ1η = A092569, ητ2 = τ2η = A117120.
({ϵ, η, τ1, ητ1}, ∘) is a Klein 4-group (C2xC2, ), 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, ).
• 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, ).
• 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, ).
# 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), (, ):

 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] ().
MERINO MAESTRE M., YURRAMENDI MENDIZABAL Y. 2014. "Lauki sareko patroi bitarren kalkulua, oinarrizko konbinatoriaren eskutik" EKAIA, 27, 237-262 ().
MERINO MAESTRE M., UNANUE GUAL I. 2018. "Lauki sareko patroien kalkulua, Polyaren teoriaren eskutik" EKAIA, 34, 289-316 ().