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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 )

See 'Index entries for fraction trees' ()

A002487 can be represented by blocks of 2m terms (m ≥ 0) in a natural way ().

• The classes of enumeration systems has been generated by means of a combinatorial procedure by taking into account the blocks, that is to say, block by block (()). More classes can be defined.
• In these enumeration systems every numerator (denominator) is a permuted sequence of A002487, where the permutation is produced at each block of 2m terms.
• Sequences num+den (= numerator+denominator) are considered. In sequences num+den each integer n > 1 appears phi(n) times (phi, Euler's totient function, A000010)). Sequence built by sums of blocks of num+den is A083329.
• numerator/num+den and denominator/num+den are enumeration systems of rationals in (0,1).
• 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.
• And, viceversa, what irreducible fraction corresponds to a given location.
Same questions can be considered for numerator, denominator, and num+den sequences. For example, a fast algorithm for A002487 has been defined ().

### Classification

#### Class 1

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

Example:

f(n)              numerator(n)
Calkin-Wilf:  f(2n) = ------  =  -----------------------------           f(2n+1) = f(n) + 1
f(n)+1      numerator(n) + denominator(n)

The four systems organized in a table

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

The numerator and denominator of four systems organized in a table

∀n > 0    numerator(2n+1) = denominator(2n) = num+den(n) = numerator(n) + denominator(n)

denominator(2n+1) =    denominator(n)       numerator(n)
------------------------------------------
numerator(n) |       Calkin-Wilf         Yu-Ting-1     |
numerator(2n) =               |                                         |
denominator(n)|       Yurramendi-2           driB       |
------------------------------------------

#### Class 2

numerator(2m+k)                      &n bsp;                 numerator(2m+k) + denominator(2m+k)
∀m ≥ 0, ∀k such that 0 ≤ k < 2m    f(2m+k) = ----------------------------             f(2m+k)+1 = -------------------------------------------------------
denominator(2m+k)                                                   denominator(2m+k)

Example:

f(2m+k)                          numerator(2m+k)
Stern-Brocot:  f(2m+1+k) = ---------------  =  --------------------------------------------------------                  f(2m+1+2m+k) = f(2m+k) + 1
f(2m+k)+1        numerator(2m+k) + denominator(2m+k)

f(2m+1+k) =    f(2m+k)/(f(2m+k)+ 1)                    1/(f(2m+k)+1)

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

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

denominator(2m+1+2m+k) =    denominator(2m+k)      numerator(2m+k)

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

### Permutations

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

##### Class 2

Relationships

∘   | A000027 A054429 A258746 A180200 A233279 A165199 A154435 A006068
---------------------------------------------------------------------------
A000027 | A000027 A054429 A258746 A180200 A233279 A165199 A154435 A006068 |
A054429 | A054429 A000027 A165199 A154435 A006068 A258746 A180200 A233279 |
A258746 | A258746 A165199 A000027 A233279 A180200 A054429 A006068 A154435 |
A180200 | A180200 A006068    a       b       c       m       n       o    |
A233279 | A233279 A154435    d       e       f       p       q       r    |
A165199 | A165199 A258746 A054429 A006068 A154435 A000027 A233279 A180200 |
A154435 | A154435 A233279    p       q       r       d       e       f    |
A006068 | A006068 A180200    m       n       o       a       b       c    |
-------------------------------------------------------------------

a <- A180200[A258746], b <- A180200[A180200], c <- A180200[A233279]
d <- A233279[A258746], e <- A233279[A180200], f <- A233279[A233279]
m <- A180200[A165199], n <- A180200[A154435], o <- A180200[A006068]
p <- A233279[A165199], q <- A233279[A154435], r <- A233279[A006068]

#### Permutations between numerator and denominator of systems

σ1 = A000027, the positive integers (identity permutation).
τ0 = A054429, the inverse permutation by blocks of 2m terms.
τ1 = A063946, τ2 = A065190.

• τ0τ0 = σ1.
({σ1, τ0}, ∘) is a cyclic group (C2, ).
• τ1τ1 = τ2τ2 = σ1.
τ0τ1 = τ1τ0 = A117120, τ0τ2 = τ2τ0 = A092569.
({σ1, τ0, τ1, τ0τ1}, ∘) is a Klein 4-group (C2xC2, ), and so is ({σ1, τ0, τ2, τ0τ2}, ∘).
• τ1τ2 = τ2τ1.
({σ1, τ0, τ1, τ2, τ0τ1, τ0τ2, τ1τ2, τ0τ1τ2}, ∘) is an elementary abelian group of order 23 (C2xC2xC2, ).
##### Class 1
• Calkin-Wilf:              A002487τ0 = , ∘τ0 = A002487

#### Permutations between systems

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

σ1 = A000027, the positive integers (identity permutation).
σ0 = A059893, the bit-reversal permutation by blocks of 2m terms.

##### Within classes

Structure of permutation system is the same in both classes.

Class 1/Class 2
 From                To Stern-Brocot/Calkin-Wilf Bird/driB HCS/Yu-Ting Yurramendi-1/2 Stern-Brocot/Calkin-Wilf σ1 σ2 σ3 σ4 Bird/driB σ2 σ1 σ4 σ3 HCS/Yu-Ting σ3' σ4' σ1 σ2' Yurramendi-1/2 σ4' σ3' σ2' σ1

σ1 = A000027

• σ2σ2 = σ'2σ'2 = σ1
σ2σ'2 = σ'2σ2
({σ1, σ2, σ'2, σ2σ'2}, ∘) is a Klein 4-group (C2xC2).
• σ3σ'3 = σ'3σ3 = σ1
σ4σ'4 = σ'4σ4 = σ1
###### Within Class 1

σ2=A258996, σ'2=A092569, σ2σ'2 = A284447,
σ3=A231551, σ'3=A231550,
σ4=A284459, σ'4=A284460.

###### Within Class 2

σ2=A258746, σ'2=A117120, σ2σ'2 = A284120,
σ3=A233279, σ'3=A233280,
σ4=A180200, σ'4=A180201.

#### Some other relationships between permutations

σ1 = A000027,
τ0 = A054429, σ0 = A059893,
τ1 = A063946, τ2 = A065190,
σ2-1 = A258996, σ'2-1 = A092569, σ2-2 = A258746, σ'2-2 = A117120 .

• σ0τ0 = τ0σ0 = A059894,
({σ1, σ0, τ0, σ0τ0}, ∘) is a Klein 4-group
• σ2-1τ0 = τ0σ2-1,          σ2-1τ1 = τ1σ2-1,         σ2-1τ2 = τ2σ2-1
σ'2-1τ0 = τ0σ'2-1 = τ2,       σ'2-1τ1 = τ1σ'2-1,         σ'2-1τ2 = τ2σ'2-1 = τ0
({σ1, τ0, τ1, τ2, σ0, σ2-1, σ'2-1, ...(*), τ0τ1τ2σ0σ2-1σ'2-1}, ∘) is also an elementary abelian group of order 26.
• σ2-2τ0 = τ0σ2-2 = A165199,    σ2-2τ1 = τ1σ2-2,         σ2-2τ2 = τ2σ2-2
σ'2-2τ0 = τ0σ'2-2 = τ1,       σ'2-2τ1 = τ1σ'2-2 = τ0,      σ'2-2τ2 = τ2σ'2-2
({σ1, τ0, τ1, τ2, σ0, σ2-2, σ'2-2, ...(*), τ0τ1τ2σ0σ2-2σ'2-2}, ∘) is an elementary abelian group of order 26, where (*) expresses all the k-combinations (1<k<6) from the set of 6 basic permutations ({τ0, τ1, τ2, σ0, σ2-2, σ'2-2}; all except σ1).
• σ2-1σ2-2 = σ2-2σ2-1,        σ2-2σ'2-1 = σ'2-1σ2-2
σ'2-1σ2-2 = σ2-2σ'2-1,       σ'2-2σ'2-1 = σ'2-1σ'2-2
({σ1, τ0, τ1, τ2, σ0, σ2-2, σ'2-2, σ2-1, σ'2-1, ...(*), σ1τ0τ1τ2σ0σ2-2σ'2-2σ2-1σ'2-1}, ∘) is an elementary abelian group of order 28.
• σ4-2σ3-2 = A064707       σ'3-2σ'4-2 = A064706

## 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 ().