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# User:Yosu Yurramendi

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

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 based on Stern's sequence ( A002487 )

See 'Index entries for fraction trees' ([6])

A002487 can be represented by blocks of 2m terms (m ≥ 0) in a natural way ([7]).

• 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 (([8])). 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).

### Classification

#### Class 1

numerator(2m+k)                                        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        |
________________________________________

#### Class 2

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)

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         |
____________________________________________

∀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          |
________________________________________

### Permutations

#### 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, [16]).
• τ1τ1 = τ2τ2 = σ1.
τ0τ1 = τ1τ0 = A117120, τ0τ2 = τ2τ0 = A092569.
({σ1, τ0, τ1, τ0τ1}, ∘) is a Klein 4-group (C2xC2, [17]), 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, [18]).

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

###### Within Class 2

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

#### Some other relationships between permutations

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

• σ0τ0 = τ0σ0 = A059894,
({σ1, σ0, τ0, σ0τ0}, ∘) is a Klein 4-group
• σ2-1τ0 = τ0σ2-1 = A165199,    σ2-1τ1 = τ1σ2-1,         σ2-1τ2 = τ2σ2-1
σ'2-1τ0 = τ0σ'2-1 = τ1,       σ'2-1τ1 = τ1σ'2-1 = τ0,      σ'2-1τ2 = τ2σ'2-1
({σ1, τ0, τ1, τ2, σ0, σ2-1, σ'2-1, ...(*), τ0τ1τ2σ0σ2-1σ'2-1}, ∘) 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, σ2-1, σ'2-1}; all except σ1).
σ2-2τ0 = τ0σ2-2,          σ2-2τ1 = τ1σ2-2,         σ2-2τ2 = τ2σ2-2
σ'2-2τ0 = τ0σ'2-2 = τ2,      σ'2-2τ1 = τ1σ'2-2,         σ'2-2τ2 = τ2σ'2-2 = τ0
({σ1, τ0, τ1, τ2, σ0, σ2-2, σ'2-2, ...(*), τ0τ1τ2σ0σ2-2σ'2-2}, ∘) is also an elementary abelian group of order 26.
• σ2-1σ2-2 = σ2-2σ2-1,       σ2-1σ'2-2 = σ'2-2σ2-1
σ'2-1σ2-2 = σ2-2σ'2-1,       σ'2-1σ'2-2 = σ'2-2σ'2-1
({σ1, τ0, τ1, τ2, σ0, σ2-1, σ'2-1, σ2-2, σ'2-2, ...(*), σ1τ0τ1τ2σ0σ2-1σ'2-1σ2-2σ'2-2}, ∘) is an elementary abelian group of order 28.
• σ4-1σ3-1 = A064707       σ'3-1σ'4-1 = 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] ([19]).
MERINO MAESTRE M., YURRAMENDI MENDIZABAL Y. 2014. "Lauki sareko patroi bitarren kalkulua, oinarrizko konbinatoriaren eskutik" EKAIA, 27, 237-262 ([20]).