Resea h A ile
Ma hema ial
Me hods in he
Applied Sienes
Reei ed XXXX
(www.in e siene.wiley.om) DOI: 10.1002/sim.0000
MOS subje lassia ion: 05B15; 13F20; 20N05; 05B25.
Coun ing and enume a ing pa ial La in
e angles by means o ompu e algeb a
sys ems and CSP sol e s
Raul M. Falon
a

,

Osa J. Falon
b
and Juan Nu~nez
b
This pap e p o ides an in-dep h analysis o how ompu e algeb a sys ems and CSP sol e s an b e used o deal wi h he
p oblem o enume a ing and dis ibu ing he se o

s
pa ial La in e angles based on
n
symb ols ao ding o hei
weigh , shap e, yp e o s u u e. The ompu a ion o Hilb e un ions and iangula sys ems o adial ideals enables us
o sol e his p oblem o all
; s ; n

6
. As a by-p o du , explii o mulas a e de e mined o he numb e o pa ial La in
e angles o weigh up o six. Fu he , in o de o illus a e he ee i eness o he ompu a ional me ho d, we o us
on he enume a ion o h ee subse s: (a) non-omp essible and egula , (b) o ally symme i, and () o ally onjuga e
o hogonal pa ial La in squa es. In pa iula , he o me enables us o enume a e he se o semine s o p oin ank up
o eigh and o p o e he exis ene o wo new ongu a ions o poin ank eigh . Finally, as an illus a i e applia ion, i
is also exp osed a me ho d o ons u o ally symme i pa ial La in squa es ha gi es ise, unde e ain ondi ions, o
new amilies o Lie pa ial quasig oup ings. Copy igh


2017 John Wiley & Sons, L d.
Keywo ds:
Pa ial La in squa e; polinomial ing; ideal; semine ; onjugay; o hogonali y.
1. In o du ion
An

s
pa ial La in e angle based on he se
[
n
℄ :=
1
; : : : ; n
g
is an

s
a ay in whih eah ell is ei he emp y o on ains
one symbol hosen om he se [
n
℄, suh ha eah symbol ou s a mos one in eah ow and in eah olumn. I s
weigh
is he numbe o non-emp y ells. This is a
La in e angle
i he e a e no emp y ells. I
=
s
=
n
, hen i is a
pa ial La in
squa e
o o de
n
(a
La in squa e
i he e a e no emp y ells). He ea e ,
R
;s ;n
and
R
;s ;n
;
m
deno e, espe i ely, he se o

s
pa ial La in e angles based on [
n
℄ and i s subse o elemen s o weigh
m
.
Coun ing, enume a ing and lassi ying La in e angles a e lassial p oblems in ombina o ial design heo y. Cu en ly, i is
known [1{4℄ he numbe o La in squa es o o de up o 11 and hei dis ibu ion in o iso opism, isomo phism and main lasses,
oge he wi h he numbe o

s
La in e angles based on [
n
℄, o

s
=
n

11 and some esul s o

6 and
s
=
n >
11
(see [5,6℄ and he e e enes he ein). Ne e heless, he equi alen p oblems o pa ial La in e angles ha e no been deal wi h
in dep h ye . Pa iula ly, by means o ompu a ional algeb ai geome y, i is known [7{9℄ he numbe o pa ial La in squa es
o o de up o six and hei dis ibu ion in o iso opism and isomo phism lasses, oge he wi h he a dinali y o
R
;s ;n
;
m
o
; s ; n

4 (see [10,11 ℄ o p e ious s udies abou how o use his ompu a ional me hod in o de o deal wi h La in squa es).
b
Faul y o Ma hema is, Depa men o Geome y and Topology, Uni e si y o Se ille, / Ta a s/n. 41012-Se illa.
a
Uni e si y o Se ille, Depa men o Applied Ma hema is I.

Co espondene o: Shool o Building Enginee ing, Uni e si y o Se ille. A da. Reina Me edes 4 A, 41012, Se ille, Spain. E-mail: a alganus.es
"This is he p e-pee e iewed e sion o he ollowing a icle: [R. M. Falcón, O. J. Falcón and J. Núñez. Coun ing and enume a ing
pa ial La in ec angles by means o compu e algeb a sys ems and CSP sol e s. Ma hema ical Me hods in he Applied Sciences
(2018). DOI: 10.1002/mma.4820], which has been published in inal o m a [h ps://doi.o g/10.1002/mma.4820]. This a icle may be
used o non-comme cial pu poses in acco dance wi h Wiley Te ms and Condi ions o Sel -A chi ing."
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
This pape p o ides an in-dep h analysis o how ompu a ional algeb ai geome y an be used o enume a e and lassi y
pa ial La in e angles ao ding no only o hei weigh , bu also o hei shape, ype and s u u e. In o de o illus a e
he ee i eness o his ompu a ional me hod, we ous on he enume a ion o (a) non-omp essible and egula , (b) o ally
symme i, and () o ally onjuga e o hogonal pa ial La in squa es. The o me enables us o deal wi h he enume a ion o
semine s (a ype o iniden s u u e in odued by Usan [12℄ as a na u al gene aliza ion o ne s), whe eas he s udy o he
o he wo ypes o pa ial La in squa es a e ela ed o algeb ai p ope ies o pa ial quasig oups (a b ie ske h o his s udy
has een ly been exposed by he au ho s in [13℄). Reall in his las ega d ha a
quasig oup
o
o de
n
[14℄ is a pai (
S;

)
o med by a ni e se
S
o
n
elemen s ha is endowed wi h a p odu

so ha , i any wo o he h ee symbols in he equa ion
a

b
=

a e gi en as elemen s o
S
, hen he hi d one is uniquely de e mined. This onep is s aigh o wa dly gene alized o
ha o
pa ial quasig oup
o o de
n
, o whih (a) he law

is a pa ial bina y ope a ion, and (b) i bo h equa ions
a

x
=
b
and
y

a
=
b
, wi h
a; b
2
S
, ha e solu ions o
x ; y
2
S
, hen bo h solu ions a e unique. The mul iplia ion able o a (pa ial)
quasig oup o o de
n
ons i u es indeed a (pa ial) La in squa e o he same o de .
B uk [15℄ in odued he onep o
o ally symme i quasig oup
as a quasig oup (
S;

) o whih he equa ion
a

b
=

emains alid unde e e y pe mu a ion o he h ee symbols
a; b ; 
2
S
. The e exis six suh pe mu a ions and eah one o
hem gi es ise o a new quasig oup, whih is said o be
onjuga e
o (
S;

). Hene, a quasig oup is o ally symme i i i s six
onjuga es oinide. I besides, he quasig oup is
idempo en
, ha is, i
a

a
=
a
, o all
a
2
S
, hen his no ion is equi alen o
ha o a
S eine iple sys em
. The dis ibu ion o o ally symme i quasig oups and S eine iple sys ems in o isomo phism
lasses is known [16,17℄ o o de s up o 10 and 19, espe i ely.
Two quasig oups o o de
n
a e said o be
o hogonal
i he jux aposi ion o hei o esponding mul iplia ion ables gi es
ise o an
n

n
a ay on aining
n
2
dis in o de ed pai s. S ein [18℄ posed he p oblem o ons u ing a quasig oup o La in
squa e ha is o hogonal o one o i s onjuga es. In his ega d, i is known [19{22℄ he exis ene o quasig oups ha a e
o hogonal o he onjuga e unde onside a ion, whih is in u n dis in om he o me , o any o de
n
62
2
;
3
;
6
g
. Muh
mo e een ly, Benne and Zhang [23℄ deal wi h La in squa es o whih eah one o hei onjuga es is o hogonal o i s
anspose. They p o ed he exis ene o suh La in squa es o all p ime powe s
n
62
2
;
3
;
5
g
. Fu he , Lindne e al. [24 ℄ oused
on idempo en La in squa es o whih hei six onjuga es a e dis in and pai wise o hogonal. They p o ed in pa iula he
exis ene o suh La in squa es o e e y o de being a p ime powe
n

8 and also o all suÆien ly la ge o de s
n
. Benne [25 ℄
es ablished
n >
5594 as an uppe bound o his las ondi ion exep possibly
n
= 6810, and enume a ed a se ies o smalle
o de s o whih hese La in squa es also exis . Fou yea s la e , he imp o ed [26℄ he p e ious uppe bound o
n >
5074. Muh
mo e een ly, Belya skaya and Popo ih [27℄ in odued he equi alen no ion o
o ally onjuga e o hogonal quasig oup
as a
quasig oup o whih i s six onjuga es a e dis in and pai wise o hogonal. They p o ed he exis ene o suh quasig oups o
any o de
n

11 ha is ela i ely p ime o 2, 3, 5, and 7. Thei mo i a ion o s udy his kind o quasig oups was mainly based
on hei applia ion in e o de e ing odes [28℄.
Sine E ans [29℄ in odued he p oblem o embedding a pa ial quasig oup o o de
n
in o a quasig oup o o de 2
n
, a wide
amoun o au ho s ha e deal wi h he embedding o dis in ypes o pa ial quasig oups; pa iula ly, ha o a pa ial o ally
symme i quasig oup in o a o ally symme i quasig oup [30 {33℄. Fu he , he o hogonali y among onjuga es o a pa ial
La in squa e was indi e ly on empla ed [34{36℄ by ousing on he exis ene o inomple e La in squa es ha a e o hogonal
o one o hei onjuga es and ha e an emp y subsqua e ha an be lled by means o a La in squa e ha is o hogonal in
u n o i s o esponding onjuga e. A mo e gene al ase was p oposed by he  s au ho [8℄, who makes use o ompu a ional
algeb ai geome y o enume a e he se o sel -o hogonal pa ial La in squa es o o de
n

4. This pape del es in o his
opi by dealing wi h he se s o pa ial La in squa es o a gi en o de o whih hei six onjuga es ei he oinide o a e all o
hem dis in and pai wise o hogonal, espe i ely. In o de o imp o e he ompu a ional eÆieny, i is p oposed o ous on
ehniques o sol e Boolean sa isabili y p oblems ins ead o hose on algeb ai geome y.
As an illus a i e applia ion o he exposed s udy, we also del e in o a een wo k de eloped by he au ho s [37℄ abou he
enume a ion o pa ial quasig oup ings o e ni e elds de i ed om pa ial La in squa es. B uk [15℄ in odued he onep
o
quasig oup ing
ela ed o a quasig oup (
S;

) as an algeb a o basis
e
a
j
a
2
S
g
o e a base eld
K
suh ha
e
a
e
b
=
e
a

b
,
o all
a; b
2
S
. This onep is s aigh o wa dly gene alized o ha o
pa ial quasig oup ing
in ase o being he pai (
S;

) a
pa ial quasig oup. In his pape , we des ibe a o ally symme i pa ial La in squa e o o de 3
n
, de i ed om a gi en pa ial
La in squa e o o de
n
, ha enables us o in odue in u n a Lie pa ial quasig oup ing o e a ni e eld o ha a e is i wo.
2Copy igh


2017 John Wiley & Sons, L d.
Ma h. Me h. Appl. Si.
2017
,001{25
P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
The pape is o ganized as ollows. Se ion 2 deals wi h some p elimina y onep s and esul s on pa ial La in squa es,
semine s and ompu a ional algeb ai geome y ha a e used h oughou ou s udy. These esul s a e implemen ed in Se ion
3 o de e mine he a dinali y o
R
;s ;n
;
m
, o all
; s ; n

6. In Se ion 4, he dis ibu ion o non-emp y ells pe ow and olumn
and he numbe o ou enes o eah symbol enable us o use ompu a ional algeb ai geome y in o de o iden i y he se
o pa ial La in e angles o a gi en shape, ype o s u u e. The dis ibu ion o
R
;s ;n
in o iso opism and main lasses is hen
de e mined o all
; s ; n

6. As a by-p odu , we es ablish explii o mulas o he numbe o pa ial La in e angles o any
o de and weigh up o six. Se ion 5 deals wi h he dis ibu ion in o main lasses o semine s o poin ank up o eigh . We
also p o e he exis ene o wo new ongu a ions o semine s wi h poin ank eigh ha omple e he lassia ion gi en by
Lyakh [38℄. In Se ion 6, we in odue a pai o se ies o bina y ons ain s ha ha a e ize, espe i ely, he se s o o ally
symme i and o ally onjuga e o hogonal pa ial La in squa es o gi en o de and weigh . Finally, Se ion 7 deals wi h an
illus a i e me hod o ons u a amily o Lie pa ial quasig oup ings om e ain o ally symme i pa ial La in squa es.
2. P elimina ies
This se ion deals wi h some basi esul s on pa ial La in e angles, semine s and ompu a ional algeb ai geome y ha a e
used h oughou he pape . Fo mo e de ails abou hese opis, we e e he eade o [12,39 ,40 ℄.
2.1. Pa ial La in e angles
An
en y
o a pa ial La in e angle
P
2 R
;s ;n
is any iple (
i ; j ; k
)
2
[
℄

[
s
℄

[
n
℄ ha is uniquely ela ed o a non-emp y
ell o
P
whih is si ua ed in he
i
h
ow and
j
h
olumn and on ains he symbol
k
. The pa ial La in e angle
P
is uniquely
de e mined by he se o all i s en ies, whih is deno ed as
E
(
P
). Thus, o ins ane, he pa ial La in squa e
P
in Figu e 1
belongs o he se
R
3
;
3
;
3;4
and has
(1
;
1
;
2)
;
(1
;
2
;
1)
;
(2
;
1
;
1)
;
(3
;
3
;
3)
g
as se o en ies.
P

2 1
1
3
Q

1
3 2
3
Figu e 1. Iso opi pa ial La in squa es in
R
3
;
3
;
3;4
.
Le
S
m
deno e he symme i g oup on
m
elemen s. An
iso opism
o
R
;s ;n
is any iple  = (
;  ; 
)
2
S

S
s

S
n
, whe e

,

and

ons i u e, espe i ely, a pe mu a ion o he ows, olumns and symbols o any pa ial La in e angle
P
2 R
;s ;n
.
This gi es ise o he
iso opi
pa ial La in e angle
P

2 R
;s ;n
, whose se o en ies is
E
(
P

) =
(

(
i
)
; 
(
j
)
; 
(
k
)) : (
i ; j ; k
)
2
E
(
P
)
g
. Thus, o ins ane, bo h pa ial La in squa es in Figu e 1a e iso opi by means o he iso opism ((123)
;
(12)
;
(13)).
Pe mu a ions among he h ee omponen s o all he en ies o a pa ial La in e angle also gi e ise o new pa ial La in
e angles. In his ega d, le

be a pe mu a ion in
S
3
. The

-onjuga e
o
P
2 R
;s ;n
is dened as he pa ial La in e angle
P

ha ing as se o en ies he se
E
(
P

) =
(
p

(1)
; p

(2)
; p

(3)
) : (
p
1
; p
2
; p
3
)
2
E
(
P
)
g
. I he pe mu a ion

p ese es he se
R
;s ;n
, hen

is said o be a
pa as ophism
. Hene, he se o pa as ophisms o
R
;s ;n
is

Id
g
i
,
s
and
n
a e pai wise dis in .

Id
;
(12)
g
i
=
s
6
=
n
.

Id
;
(13)
g
i
=
n
6
=
s
.

Id
;
(23)
g
i
s
=
n
6
=
.

S
3
i
=
s
=
n
.
The e a e, he e o e, six onjuga es:
P
Id
=
P
,
P
(12)
=
P
,
P
(13)
,
P
(23)
,
P
(123)
= (
P
(23)
)
and
P
(132)
= (
P
(13)
)
; whe e
deno es
he anspose o he o esponding pa ial La in e angle. Figu e 2shows, o ins ane, a pa ial La in squa e
P
whose six
onjuga es a e pai wise dis in . The pa ial La in squa e
P
ha is shown in Figu e 1is, howe e , an example o whih all i s
six onjuga es oinide. Suh a pa ial La in squa e is said o be
o ally symme i
. He ea e , we deno e espe i ely as TS
n
and TS
n
;
m
he se o o ally symme i pa ial La in squa es o o de
n
and i s subse o pa ial La in squa es o weigh
m
.
Ma h. Me h. Appl. Si.
2017, 00 1{25 Copy igh


2017 John Wiley & Sons, L d.
3
P epa ed using mmaau h.ls
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
P

1 2
3
1
P

1
2 3
1
P
(13)

1 3
1
2
P
(23)

1 2
2
3
P
(123)

1 3
2
2
P
(132)

1
1 2
3
Figu e 2. Pa ial La in squa e in
R
3
;
3
;
3;4
and i s onjuga es.
Two pa ial La in e angles a e said o be
pa a opi
i one o hem is iso opi o a onjuga e o he o he . To be iso opi,
pa as ophi o pa a opi a e equi alene ela ions among pa ial La in e angles. They make possible he espe i e dis ibu ion
o pa ial La in e angles in o
iso opism
,
pa as ophism
and
main
lasses.
A pa ial La in squa e
P
o o de
n
is said o be
non-omp essible
i his does no on ain emp y ows o emp y olumns, o i
all he
n
symbols appea as en ies in
E
(
P
). This is said o be
egula
i : (a) he e does no exis a ell ha is, simul aneously,
he only non-emp y ell in i s ow and i s olumn, and (b) any ow o olumn wi h exa ly one non-emp y ell on ains a symbol
ha appea s a leas wie in
E
(
P
). Thus, o ins ane, he pa ial La in squa e
P
in Figu e 2is non-omp essible. Ne e heless,
i is no egula , beause: (a) bo h i s hi d ow and i s hi d olumn ha e exa ly one non-emp y ell, whih is ommon o bo h
o hem, and (b) i s seond ow on ains exa ly one non-emp y ell, bu he symbol he ein only appea s one in
P
.
Two pa ial La in squa es o o de
n
,
P
= (
p
i j
) and
Q
= (
q
i j
), a e said o be
o hogonal
i all he o de ed pai s on non-
emp y en ies ha a e ob ained when bo h a ays a e supe imposed a e dis in . Equi alen ly, gi en
i ; i
0
; j ; j
0
2
[
n
℄ suh ha
p
i j
=
p
i
0
j
0
2
[
n
℄, hen
q
i j
and
q
i
0
j
0
a e no he same symbol o [
n
℄. Thus, o ins ane, he pa ial La in squa es
P
and
P
(13)
in
Figu e 2a e o hogonal, bu he pa ial La in squa es
P
and
P
(12)
in he same gu e a e no . Now, le us onside a non- i ial
pe mu a ion

2
S
3
n
Id
g
. A pa ial La in squa e
P
2 R
n;n ;n
is said o be

-o hogonal
i i is o hogonal o i s

-onjuga e.
This is
sel -o hogonal
i

= (12). Thus, o ins ane, he pa ial La in squa e
P
(23)
in Figu e 2is sel -o hogonal. Fu he , we
say ha a pa ial La in squa e is
o ally onjuga e o hogonal
i i s six onjuga es a e dis in and pai wise o hogonal. This is
he ase, o ins ane, o he pa ial La in squa e in Figu e 3. F om he e on, he se o o ally onjuga e o hogonal pa ial La in
squa es o o de
n
and i s subse o pa ial La in squa es o weigh
m
a e espe i ely deno ed as TCO
n
and TCO
n
;
m
.
P

3
2
1 3
P

1
3
3 2
P
(13)

3
2
3 1
P
(23)

3
3
1 2
P
(123)

1
3
3 2
P
(132)

3
3
2 1
Figu e 3. To ally onjuga e o hogonal pa ial La in squa e in
R
3
;
3
;
3;4
.
2.2. Semine s
Ba es [41℄ dened a
hal ne
as an inidene s u u e o poin s and lines suh ha : (a) he e exis h ee dis in
pa allel lasses
o lines, (b) e e y poin is on a mos one line o eah lass, and () any wo lines belonging o dis in lasses mee in a mos
one poin . The numbe o poin s ons i u es he
poin ank
o a hal ne . Two hal ne s a e in he same
isomo phism lass
i
he e exis s a pe mu a ion among he poin s ha p ese es ollinea i y in eah pa allel lass. I his happens a e elabeling hei
pa allel lasses, hen hey a e in he same
main lass
. Cu en ly, he dis ibu ion o hal ne s in o isomo phism and main lasses
is only pa ially known o ne s and, o a muh lesse ex en , semine s.
B uk [42℄ dened a
ne
o o de
n
as a hal ne o
n
2
poin s and 3
n
lines in whih e e y poin is on exa ly one line o eah
pa allel lass, any wo lines om dis in pa allel lasses mee in exa ly one poin and he e exis s a leas one line wi h exa ly
n
dis in poin s. Hene, e e y line on ains
n
poin s and e e y pa allel lass is o med by
n
lines. Mo e een ly and mo i a ed by
i s applia ion in oding heo y, Usan [12℄ in odued he onep o
semine
as a hal ne in whih e e y poin is on exa ly one
line o eah pa allel lass and any wo lines mee in a mos one poin . Unlike ne s, he lines o a semine an on ain die en
numbe s o poin s and i s pa allel lasses an ha e die en numbe s o lines. The
L
-o de
o a semine is he maximum numbe
o lines in a pa allel lass. I all he lines ha e he same numbe
n
o poin s, hen all he pa allel lasses ha e he same numbe
m
o lines. In his ase, he semine is said o be
n
- egula
. I , u he mo e,
m
=
n
, hen i is a ne o o de
n
.
4Copy igh


2017 John Wiley & Sons, L d.
Ma h. Me h. Appl. Si.
2017
,001{25
P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes

1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
Figu e 4. Ne iden ied wi h a La in squa e o o de 4.
E e y ne o o de
n
an be iden ied wi h a La in squa e o he same o de . The poin s and pa allel lasses o he ne a e
espe i ely iden ied wi h he ells o he La in squa e and i s se s o ells sha ing he same ow, olumn o symbol (see Figu e
4). In addi ion, S ojako i and Usan [43 ℄ p o ed ha e e y semine o
L
-o de
n
an be iden ied wi h a non-omp essible egula
pa ial La in squa e o o de
n
in a simila way ha ne s do wi h La in squa es. In his ase, he poin s o he semine a e iden ied
wi h he non-emp y ells o he pa ial La in squa e (see Figu e 5). As a onsequene, he dis ibu ion o ne s and semine s
in o isomo phism and main lasses esul s, espe i ely, om he equi alen dis ibu ion o La in squa es and non-omp essible
egula pa ial La in squa es in o iso opism and main lasses.

1 2
1 2
2
Figu e 5. Semine iden ied wi h a pa ial La in squa e o o de 4 and weigh 5.
Ha el [44℄ dened a
ongu a ion
as a semine on aining a leas ou poin s suh ha e e y line on ains a leas wo
poin s and any wo poin s
P
and
Q
o he semine a e
onne ed
, ha is o say, he e exis s a sequene o poin s and lines,
P
0
; l
0
; P
1
; l
1
;:::;P
m
, suh ha
P
0
=
P
,
P
m
=
Q
and eah pai o poin s
P
i

1
and
P
i
a e on he line
l
i

1
, o all
i

m
. Ha el
de e mined he main lasses o hose ongu a ions wi h poin ank up o se en and, sho ly a e , Lyakh [38℄ ga e a lassia ion
o hose ongu a ions wi h poin ank eigh .
2.3. Compu a ional algeb ai geome y
Le
X
and
K
[
X
℄ espe i ely be he o de ed se o
n
a iables
x
1
;:::;x
n
g
and he ela ed mul i a ia e polynomial ing
K
[
x
1
;:::;x
n
℄ o e a base eld
K
. The
lass
o a polynomial
p
2
K
[
X
℄ is he minimum
i

n
suh ha
p
2
K
[
x
1
;:::;x
i
℄. A
iangula sys em
in
K
[
X
℄ is a ni e o de ed se o polynomials
p
1
;:::;p
m
g 
K
[
X
℄ suh ha he lass o
p
i
is less han he
lass o
p
i
+1
, o all
i < m
. An
ideal
o polynomials in
K
[
X
℄ is any subse
I

K
[
X
℄ suh ha 0
2
I
;
p
+
q
2
I
, o all
p ; q
2
I
;
and
p q
2
I
o all
p
2
I
and
q
2
K
[
X
℄. A
subideal
o
I
is any subse
J

I
ha is also an ideal in
K
[
X
℄. The ideal
gene a ed by
a ni e se o polynomials
p
1
;:::;p
m
g 
K
[
X
℄ is dened as he se
q
1
p
1
+
:::
+
q
n
p
n
:
q
1
;:::;q
n
2
K
[
X
℄
g
. The
aÆne a ie y
V
(
I
) is he se o poin s in
K
n
ha a e ze os o all he polynomials in
I
. I his is ni e, hen he ideal
I
is
ze o-dimensional
. I is
adial
i i on ains all he polynomials
p
2
K
[
X
℄ so ha
p
m
2
I
o some na u al
m
.
A
e m o de
on he se o monomials o
K
[
X
℄ is a mul iplia i e well-o de ing whose smalles elemen is he ons an
monomial 1. Thus, o ins ane, he
lexiog aphi
e m o de
<
lex
is dened so ha , gi en wo monomials
X
a
=
x
a
1
1
: : : x
a
n
n
and
X
b
=
x
b
1
1
: : : x
b
n
n
, one has ha
X
a
<
lex
X
b
i he e exis s a na u al
m

n
suh ha
a
i
=
b
i
o all
i

m
and
a
m
< b
m
. The
la ges monomial o a polynomial wi h espe o a e m o de is i s
leading monomial
. The
ini ial ideal
o an ideal
I

K
[
X
℄
is he ideal gene a ed by he leading monomials o he non-ze o polynomials o
I
. Any subse
G

I
whose leading monomials
gene a e his ini ial ideal is alled a
G obne basis
o
I
wi h espe o he unde lying e m o de . Any monomial o
I
ha is
no on ained in i s ini ial ideal is alled
s anda d
. Rega dless o he monomial e m o de ing, i he ideal
I
is ze o-dimensional
Ma h. Me h. Appl. Si.
2017, 00 1{25 Copy igh


2017 John Wiley & Sons, L d.
5
P epa ed using mmaau h.ls

Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
and adial, hen he numbe o s anda d monomials in
I
oinides wi h he K ull dimension o he quo ien ing
K
[
X
℄
=I
and
wi h he a dinali y o
V
(
I
). This is ob ained by means o he
Hilbe un ion
, whih maps eah non-nega i e in ege
m
on o
HF
K
[
X
℄
=I
(
m
) = dim
K
(
K
[
X
℄
m
=
(
K
[
X
℄
m
I
)). He e,
K
[
X
℄
m
deno es he se o homogeneous polynomials in
K
[
X
℄ o deg ee
m
and
HF
K
[
X
℄
=I
(
m
) oinides wi h he numbe o s anda d monomials in
I
o deg ee
m
. The p oblem o ompu ing Hilbe un ions is
NP-omple e [45℄. I s ompu a ion is based on ha o a G obne basis o he ideal, whose omplexi y in ase o dealing wi h a
ze o-dimensional ideal is
d
O
(
n
)
[46℄, whe e
d
is he maximal deg ee o he polynomials and
n
is he numbe o a iables.
The nex esul india es how ompu a ional algeb ai geome y an be used o enume a e and oun he pa ial La in
e angles in he se
R
;s ;n
. He ea e , he se o a iables and he base eld o he polynomial ing o be onside ed a e,
espe i ely,
X
=
x
111
;:::;x
s n
g
and he ni e eld
F
2
.
Theo em 2.1 ( [8℄)
The se
R
;s ;n
is iden ied wi h he se o ze os o he ze o-dimensional adial ideal in
F
2
[
X
℄
I
;s ;n
:=
h
x
ijk
x
i
0
j k
; x
ijk
x
i j
0
k
; x
ijk
x
ijk
0
:
i ; i
0

;
j ; j
0

s
;
k ; k
0

n
i
:
Besides,
jR
;s ;n
;
m
j
= HF
F
2
[
X
℄
=I
;s ;n
(
m
)
;
o all
m

0
;
and
jR
;s ;n
j
= dim
F
2
(
F
2
[
X
℄
=I
;s ;n
)
:
The p oo o Theo em 2.1 is based on he a ha e e y s anda d monomial
x
a
111
111
:::x
a
s n
s n
o he ideal
I
;s ;n
an be iden ied
wi h a pa ial La in e angle in
R
;s ;n
wi h se o en ies
(
i ; j ; k
)
2
[
℄

[
s
℄

[
n
℄ :
a
ijk
= 1
g
. Pa iula ly, he p esene o he
monomial
x
ijk
x
i
0
j k
as gene a o o he ideal
I
;s ;n
in ol es he non-exis ene o he symbol
k
wie in he
j
h
olumn; ha o
x
ijk
x
i j
0
k
in ol es he non-exis ene o he symbol
k
wie in he
i
h
ow; and ha o
x
ijk
x
ijk
0
in ol es he non-exis ene o
wo dis in symbols in he ell (
i ; j
). Based on his esul , he speialized algo i hm des ibed by Dikens ein and Tobis [47℄
was implemen ed in [8℄ o ompu ing he a dinali y o
R
;s ;n
;
m
, o all
; s ; n

4. Fo highe o de s, howe e , he equi ed
ompu a ional os u ned ou o be exessi e due o la ge memo y s o age equi emen s. This os is only due o he ompu a ion
o he o esponding Hilbe un ion, beause he se o gene a o s o
I
;s ;n
ons i u es i sel a lexiog aphi G obne basis o he
ideal. To edue i , an al e na i e p oedu e is in odued in he nex se ion. This is based on he simila i y ha exis s among
hose gene a o s in
I
;s ;n
ha o espond o dis in ows in a pa ial La in e angle. A p elimina y e sion o his p oedu e was
exposed in [9℄, whe e he a dinali y o
R
;s ;n
was ompu ed o all
; s ; n

6. Fo a be e unde s anding o his p oedu e, he
o esponding ompu a ion o
jR
3
;
3
;
3;2
j
is illus a ed in Example 1.
3. An al e na i e p o edu e o ompu e
jR
;s ;n
j
Fo eah posi i e in ege
i

we dene he ze o-dimensional subideal
I
(
i
)
;s ;n
:=
h
x
ijk
x
i j
0
k
; x
ijk
x
ijk
0
:
j ; j
0

s
;
k ; k
0

n
i 
I
;s ;n
:
The e exis dis in algo i hms [48{50℄ ha enable us o deompose he ze o-dimensional ideal
I
(1)
;s ;n
in o a ni e se
J
1
;
1
;:::;J
1
;
g
o subideals gene a ed by iangula sys ems and whose aÆne a ie ies ons i u e a pa i ion o
V
(
I
(1)
;s ;n
). The
omplexi y o his ompu a ion in he men ioned algo i hms is polynomial one a lexiog aphi G obne basis o he ideal is
known. This is ou ase, beause he se o gene a o s o
I
(1)
;s ;n
ons i u es i sel one suh a basis. Now, o eah
i >
1 and
l

,
le
J
i ;l
be he subideal o
I
(
i
)
;s ;n
whose gene a o s oinide wi h hose o
J
1
;l
a e eplaing eah a iable
x
1
j k
by
x
ijk
. Fo eah
uple (
1
;:::;
)
2
[
℄
we dene he ideal
K
1
;::: ;
:=
J
1
;
1
+
: : :
+
J
;
+
h
x
ijk
x
i
0
j k
:
i ; i
0

;
j

s
;
k

n
i
:
(1)
The iangula i y o he unde lying sys ems in ol es eah subideal
J
i ;
j
o ha e a leas one gene a o o he o m
x
i j
0
k
o
x
i j
0
k

1. The numbe o gene a o s o he seond o m in he ideal
K
1
;::: ;
ons i u es he minimum numbe o en ies in a
pa ial La in e angle ha is iden ied wi h a poin in
V
(
K
1
;::: ;
). We deno e his numbe by
m
1
;:::;
.
6Copy igh


2017 John Wiley & Sons, L d.
Ma h. Me h. Appl. Si.
2017
,001{25
P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
P op osi ion 3.1
Le
m
be a non-nega i e in ege . Then
HF
F
2
[
X
℄
=I
;s ;n
(
m
) =
X
(
1
;:::;
)
2
[
℄
m
1
;:::;

m
HF
F
2
[
X
℄
=K
1
;:::;
(
m

m
1
;::: ;
)
:
P o o .
Le
X
a
=
x
a
111
111
: : : x
a
s n
s n
be a s anda d monomial o deg ee
m
in
I
;s ;n
. Sine he ideals des ibed in (1) ons i u e a
pa i ion o he aÆne a ie y
V
(
I
;s ;n
), he e exis s exa ly one ideal
K
1
;:::;
ha on ains he poin (
a
111
;:::;a
s n
)
2 V
(
I
;s ;n
).
The esul ollows hen om he a ha he monomial
X
a
is uniquely ela ed o he s anda d monomial
x
a
0
111
111
:::x
a
0
s n
s n
o deg ee
m

m
1
;::: ;
in
K
1
;::: ;
, whe e
a
0
ijk
= 0 i
x
ijk

1 is a gene a o o
K
1
;::: ;
and
a
0
ijk
=
a
ijk
, o he wise.
2
The smalle numbe o a iables ha a e equi ed o ompu e eah addend in P oposi ion 3.1, oge he wi h he iangula i y
o he in ol ed sys em and he possible pa allel ompu a ion o de e mine dis in addends a he same ime, edue he unning
ime and os o ompu a ion o HF
F
2
[
X
℄
=I
;s ;n
(
m
) in ompa ison wi h Theo em 2.1. Mo eo e , we do no need o ompu e all
hese addends, beause HF
F
2
[
X
℄
=K
1
;:::;
(
m
) = HF
F
2
[
X
℄
=K

(1)
;:::;

(
)
(
m
), o all (
1
;:::;
)
2
[
℄
,
m

0 and

2
S
.
Example 3.2
The ideal
I
(1)
3
;
3
;
3
ela ed o he  s ow o a pa ial La in squa e o o de
3
an be deomposed in o he nex six
disjoin subideals
i)
J
1
;
1
=
I
(1)
3
;
3
;
3
+
h
x
111
; x
121
; x
131
i
.
ii)
J
1
;
2
=
I
(1)
3
;
3
;
3
+
h
x
111
; x
121
; x
131

1
; x
132
; x
133
i
.
iii)
J
1
;
3
=
I
(1)
3
;
3
;
3
+
h
x
111
; x
121

1
; x
122
; x
123
; x
131
i
.
i )
J
1
;
4
=
I
(1)
3
;
3
;
3
+
h
x
111

1
; x
112
; x
113
; x
121
; x
122
; x
131
; x
132
i
.
)
J
1
;
5
=
I
(1)
3
;
3
;
3
+
h
x
111

1
; x
112
; x
113
; x
121
; x
122
; x
131
; x
132

1
; x
133
i
.
i)
J
1
;
6
=
I
(1)
3
;
3
;
3
+
h
x
111

1
; x
112
; x
113
; x
121
; x
122

1
; x
123
; x
131
; x
132
i
.
Pa ial La in squa es o o de
3
a e hen dis ibu ed as poin s o
1.
V
(
J
1
;
1
)
i hey do no on ain he symbol
1
in hei  s ow.
2.
V
(
J
1
;
2
)
i hey on ain he symbol
1
in he ell
(1
;
3)
.
3.
V
(
J
1
;
3
)
i hey on ain he symbol
1
in he ell
(1
;
2)
.
4.
V
(
J
1
;
4
)
i hey on ain he symbol
1
in he ell
(1
;
1)
bu do no on ain he symbol
2
in hei  s ow.
5.
V
(
J
1
;
5
)
i hey on ain he symbol
1
in he ell
(1
;
1)
and he symbol
2
in he ell
(1
;
3)
.
6.
V
(
J
1
;
6
)
i hey on ain he symbol
1
in he ell
(1
;
1)
and he symbol
2
in he ell
(1
;
2)
.
Fo eah iple
(
1
;
2
;
3
)
2
[6℄
3
, we onside he ideal
K
1
;
2
;
3
=
J
1
;
1
+
J
2
;
2
+
J
3
;
3
+
h
x
ijk
x
i
0
j k
:
i ; i
0
; j ; k

3
i
:
The alues o
HF
F
2
[
X
℄
=K
1
;
2
;
3
a e exposed in Table 1.
Le
m
1
;
2
;
3
be he numbe o gene a o s o he o m
x
ijk

1
in he ideal
K
1
;
2
;
3
. Thus, o ins ane, e e y poin o he aÆne
a ie y
V
(
K
6
;
3
;
2
)
is uniquely ela ed o a pa ial La in squa e o o de
3
and weigh a leas
m
6
;
3
;
2
= 4
. This las alue holds om
he a ha he se o en ies o any suh a pa ial La in squa e always on ains he subse
(1
;
1
;
1)
;
(1
;
2
;
2)
;
(2
;
2
;
1)
;
(3
;
3
;
1)
g
.
F om P oposi ion 3.1, we ha e, o example, ha
jR
3
;
3
;
3:2
j
= HF
F
2
[
X
℄
=K
1
;
1
;
1
(2) + 3 HF
F
2
[
X
℄
=K
1
;
1
;
2
(1) + 3 HF
F
2
[
X
℄
=K
1
;
1
;
3
(1) + 3 HF
F
2
[
X
℄
=K
1
;
1
;
4
(1) + 3 HF
F
2
[
X
℄
=K
1
;
1
;
5
(0)+
3 HF
F
2
[
X
℄
=K
1
;
1
;
6
(0) + 6 HF
F
2
[
X
℄
=K
1
;
2
;
3
(0) + 6 HF
F
2
[
X
℄
=K
1
;
2
;
4
(0) + 6 HF
F
2
[
X
℄
=K
1
;
3
;
4
(0) = 270
:

Ma h. Me h. Appl. Si.
2017, 00 1{25 Copy igh


2017 John Wiley & Sons, L d.
7
P epa ed using mmaau h.ls
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
Table 1.
Hilbe un ions ela ed o he se
R
3
;
3
;
3
.
HF
F
2
[
X
℄
=K
1
;
2
;
3
(
m
)
1
:
2
:
3
m
1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.2.3 1.2.4 1.2.5 1.2.6 1.3.4 1.3.5 1.3.6 2.3.4 2.3.5 2.3.6
01111111111111111
1 18 16 16 14 11 11 14 12 10 9 12 9 10 10 8 8
2 108 84 84 62 36 36 64 45 29 24 45 24 29 32 19 19
3 264 176 176 104 42 42 116 63 29 23 63 23 29 38 16 16
4 270 150 150 66 18 18 84 32 11 8 32 8 11 16 5 5
5 108 48 48 12 2 2 24 5 1 1 5 1 1 2 1 1
612440002000000000
This ompu a ional algeb ai me hod has been implemen ed in he p oedu e
PLR
o he lib a y
pls.lib
, a ailable online
on
h p://pe sonales.us.es/ au algan/LS/pls.lib
, o he open ompu e algeb a sys em o polynomial ompu a ions
Singula [51℄. The o e ness and e mina ion o his p oedu e a e based on hose o he algo i hms des ibed in [47,48,50℄
o ompu ing Hilbe un ions. In o de o es i s eÆieny, we ha e  s ly heked he known a dinali y o
R
;s ;n
;
m
, o all
; s ; n

4 (see Table 2), whih was al eady ompu ed in [8℄. In he same ompu e sys em, an
In el Co e i7-2600 CPU (8
o es), wi h a 3.4 GHz p oesso and 16 GB o RAM
, he maximum unning ime de eases om 50 seonds in [8℄ o less han
1 seond. This o esponds o he ompu a ion o he se ies
jR
4
;
4
;
4;
m
j
. The p oedu e has hen been applied o ompu ing in
Tables 3{5 he es o ases so ha

s

n

6. The unning ime anges he e om less han 1 seond o 32 hou s. This
maximum unning ime o esponds o he ompu a ion o he se ies
jR
6
;
6
;
6;
m
j
, o whih 2,3 GB o RAM is equi ed. Fo highe
o de s, he  s se ies whose ompu a ion u ned ou o be exessi e o ou ompu e sys em due o la ge memo y s o age
equi emen s was
jR
6
;
7
;
7;
m
j
. In o de o imp o e he eÆieny o his ompu a ional algeb ai me hod, we p opose in he nex
se ion o impose some ex a algeb ai ondi ions o ou base ideal. They a e e e ed o he dis ibu ion o non-emp y ells pe
ow and olumn in a pa ial La in e angle and o he numbe o ou enes o eah symbol.
Table 2.
Dis ibu ion o
R
;s ;n
ao ding o he weigh , o

s

n

4.
jR
;s ;n
;
m
j
:s :n
m
1.1.1 1.1.2 1.1.3 1.1.4 1.2.2 1.2.3 1.2.4 1.3.3 1.3.4 1.4.4 2.2.2 2.2.3 2.2.4 2.3.3 2.3.4 2.4.4 3.3.3 3.3.4 3.4.4 4.4.4
0111111111111111 1 1 1 1 1
1 1 2 3 4 4 6 8 9 12 16 8 12 16 18 24 32 27 36 48 64
2 2 6 12 18 36 72 16 42 80 108 204 384 270 504 936 1728
3 6 24 96 8 48 144 264 768 2208 1278 3552 9696 25920
4 24 2 18 84 270 1332 6504 3078 13716 58752 239760
5 108 1008 9792 3834 29808 216864 1437696
6 12 264 7104 2412 36216 494064 5728896
7 2112 756 23760 691200 15326208
8 216 108 7776 581688 27534816
9 12 1056 283584 32971008
10 75744 25941504
11 10368 13153536
12 576 4215744
13 847872
14 110592
15 9216
16 576
To al 2 3 4 5 7 13 21 34 73 209 35 121 325 781 3601 28353 11776 116425 2423521 127545137
4. Shap e, yp e and s u u e o pa ial La in e angles
The
shape
o a pa ial La in e angle
P
= (
p
i j
)
2 R
;s ;n
is dened as he

s
bina y a ay
B
P
= (
b
i j
) suh ha
b
i j
= 1 i
(
i ; j ; p
i j
)
2
E
(
P
) and 0, o he wise. Le
i
, 
j
and s
k
espe i ely be he numbe o lled ells in he
i
h
ow and
j
h
olumn o
P
and he numbe o ou enes o he symbol
k
in
P
. Ao ding o he e minology exposed by Keedwell [52℄ and gene alized by
Bean e al. [53℄, he uples
R
= (
1
;:::;
),
C
= (
1
;:::;

s
) and
S
= (s
1
;:::;
s
n
) de e mine, espe i ely, he
ow
,
olumn
and
8Copy igh


2017 John Wiley & Sons, L d.
Ma h. Me h. Appl. Si.
2017
,001{25
P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
Table 3.
Dis ibu ion o
R
;s ;
5
ao ding o he weigh , o

s

5.
jR
;s ;
5;
m
j
:s :
5
m
1.1.5 1.2.5 1.3.5 1.4.5 1.5.5 2.2.5 2.3.5 2.4.5 2.5.5 3.3.5 3.4.5 3.5.5 4.4.5 4.5.5 5.5.5
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 5 10 15 20 25 20 30 40 50 45 60 75 80 100 125
2 20 60 120 200 130 330 620 1000 810 1500 2400 2760 4400 7000
3 60 240 600 320 1680 4800 10400 7590 20520 43200 54240 112800 233000
4 120 600 260 4140 20040 61400 40500 169920 486000 676200 1881600 5159000
5 120 4680 45600 211440 126900 891360 3594960 5641920 21612480 80602200
6 1920 54480 421200 232680 3018000 17930400 32423520 176546400 920160000
7 30720 465600 240840 6605280 60912000 130248960 1045147200 7845192000
8 6360 262200 128520 9224280 140826600 367731360 4530640800 50648616000
9 63600 27480 7983840 219307800 728440320 14444083200 249687408000
10 5280 4063680 225419040 1004380800 33852910080 944069668800
11 1100160 148010400 950238720 58065734400 2741210616000
12 120960 59047200 603722880 72278294400 6104066712000
13 13284000 249580800 64484985600 10385299320000
14 1512000 63884160 40544726400 13420351008000
15 66240 9216000 17571260160 13065814483200
16 590400 5099169600 9486099648000
17 953107200 5073056640000
18 108288000 1970474400000
19 6681600 547608096000
20 161280 107330054400
21 14667552000
22 1388160000
23 91008000
24 4032000
25 161280
To al 6 31 136 501 1546 731 12781 162661 1502171 805366 33199561 890442316 4146833121 313185347701 64170718937006
Table 4.
Dis ibu ion o
R
;s ;
6
ao ding o he weigh , o

s

6 (I).
jR
;s ;
6;
m
j
:s :
6
m
1.1.6 1.2.6 1.3.6 1.4.6 1.5.6 1.6.6 2.2.6 2.3.6 2.4.6 2.5.6 2.6.6 3.3.6 3.4.6 3.5.6 3.6.6
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 6 12 18 24 30 36 24 36 48 60 72 54 72 90 108
2 30 90 180 300 450 192 486 912 1470 2160 1188 2196 3510 5130
3 120 480 1200 2400 600 3120 8880 19200 35400 13896 37344 78360 141840
4 360 1800 5400 630 9990 48060 146700 349650 94770 392580 1115100 2547450
5 720 4320 15120 146880 678240 2168640 389340 2676240 10667160 31419360
6 720 8520 245760 1899600 8546880 961380 12082680 70540800 274470480
7 204480 3139200 21211200 1375920 36270720 326808000 1727352000
8 65160 2881800 32189400 1038960 71633160 1064140200 7893282600
9 1303200 28267200 317760 90585600 2422568400 26212965600
10 222480 13063680 69603840 3803369040 62938898640
11 2669760 29255040 4021099200 108045861120
12 190800 5112000 2756361600 130246779600
13 1152144000 107367120000
14 262828800 58252478400
15 24791040 19683613440
16 3828798720
17 384652800
18 15321600
To al 7 43 229 1045 4051 13327 1447 37273 720181 10291951 108694843 4193269 317651473 15916515301 526905708889
symbol
ypes o
P
. The
ype
o
P
is hen dened as he iple (
R ; C; S
). Thus, o ins ane, he ype o he pa ial La in squa e
o Figu e 5is ((2
;
2
;
1
;
0)
;
(2
;
1
;
1
;
1)
;
(2
;
3
;
0
;
0)). He ea e , he se o pa ial La in e angles o ype (
R ; C; S
) is deno ed by
R
R;C;S
.
Le
T
n;m
be he se o
n
- uples
T
= (
1
;:::;
n
) o
weigh
P
i

n
i
=
m
whose omponen s a e non-nega i e in ege s. The
onjuga e
o
T
is he uple
T

= (

1
;:::;

m
), whe e eah

i
is he numbe o posi i e in ege s
j

n
suh ha
j

i
. I
T
= (
1
;:::;
n
)
2 T
n;m
is ob ained a e a de easing ea angemen o he omponen s o
T
, hen
T
is said o be
majo ized
by
a seond uple
T
0
= (
0
1
;:::;
0
n
)
2 T
n;m
i
P
i

j
i

P
i

j
0
i
, o all
j

n
. This gi es ise o he so-alled
dominane o de

on
T
n;m
[54℄.
Theo em 4.1
Le
(
R ; C; S
)
2 T
;m
 T
s ;m
 T
n;m
. The se
R
R;C;S
is non-emp y only i
C

R

,
S

C

and
R

S

.
Ma h. Me h. Appl. Si.
2017, 00 1{25 Copy igh


2017 John Wiley & Sons, L d.
9
P epa ed using mmaau h.ls
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
S3S4,1 S4,2 S4,3 S4,4 S5,1 S5,2
S5,3 S5,4 S5,5 S5,6 S5,7 S6,1 S6,2
S6,3 S6,4 S6,5 S6,6 S6,7 S6,8 S6,9
S6,10 S6,11 S6,12 S6,13 S6,14 S6,15 S6,16
S6,17 S6,18 S6,19 S6,20 S6,21 S6,22 S6,23
S6,24 S6,25 S6,26 S6,27 S6,28 S6,29 S6,30
S6,31 S6,32 S6,33 S6,34 S6,35 S6,36 S6,37
S6,38 S6,39 S6,40 S6,41 S6,42 S6,43 S6,44
S6,45 S6,46 S6,47 S6,48 S6,49 S6,50 S6,51
S6,52 S6,53 S6,54 S6,55
Figu e 7. Classia ion o semine s wi h poin ank up o six.
16 Copy igh


2017 John Wiley & Sons, L d.
Ma h. Me h. Appl. Si.
2017
,001{25
P epa ed using mmaau h.ls

R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
Table 7.
Dis ibu ion in o main lasses o he se
R
eg
R;C;S
.
m z
R
z
C
z
S

eg
MC
3 21 21 21 1 1
4 2
2
2
2
2
2
2 1
21
2
4 1
1
4
24 1
21
2
21
2
4 1
5 32 2
2
1 2
2
1 4 1
21
3
12 1
31
2
31
2
2
2
1 4 1
2
2
1 2
2
1 8 1
2
2
1 2
2
1 2
2
1 32 2
21
3
24 1
6 42 2
2
1
2
2
2
1
2
8 1
3
2
2
3
2
3
12 1
2
2
1
2
36 2
21
4
144 1
1
6
720 1
2
2
1
2
2
2
1
2
48 2
21
4
48 1
41
2
2
2
1
2
2
2
1
2
16 1
321 321 321 1 1
31
3
6 1
2
3
12 2
2
2
1
2
20 4
21
4
24 1
2
3
2
3
36 1
2
2
1
2
120 5
21
4
288 2
2
2
1
2
2
2
1
2
160 4
2
3
2
3
2
3
144 2
31
3
72 1
2
2
1
2
432 4
21
4
1,296 2
1
6
4,320 1
31
3
31
3
36 1
2
2
1
2
144 2
2
2
1
2
2
2
1
2
624 7
21
4
288 1
2
2
1
2
2
2
1
2
2
2
1
2
160 3
7 43 2
3
1 2
3
1 54 2
2
2
1
3
144 2
21
5
360 1
2
2
1
3
2
2
1
3
144 1
421 321
2
321
2
4 1
2
3
1 36 3
2
2
1
3
48 2
31
4
2
3
1 144 1
2
3
1 2
3
1 162 4
2
2
1
3
360 5
21
5
360 1
2
2
1
3
2
2
1
3
144 1
3
2
1 32
2
32
2
4 1
321
2
12 2
31
4
48 1
2
3
1 72 3
2
2
1
3
192 4
21
5
480 1
321
2
321
2
24 2
31
4
48 1
2
3
1 120 5
2
2
1
3
144 3
2
3
1 2
3
1 612 6
m z
R
z
C
z
S

eg
MC
7 3
2
1 2
3
1 2
2
1
3
1,008 7
21
5
720 1
2
2
1
3
2
2
1
3
288 1
32
2
32
2
32
2
16 3
321
2
48 5
31
4
144 2
2
3
1 192 7
2
2
1
3
720 12
21
5
2,640 5
1
7
10,080 1
321
2
321
2
112 9
31
4
192 2
2
3
1 456 19
2
2
1
3
816 18
21
5
480 1
31
4
2
3
1 1,008 4
2
2
1
3
288 1
2
3
1 2
3
1 1,692 16
2
2
1
3
3,744 26
21
5
6,480 5
2
2
1
3
2
2
1
3
2,592 6
321
2
321
2
321
2
144 5
2
3
1 684 18
2
2
1
3
264 5
31
4
2
3
1 432 2
2
3
1 2
3
1 2,556 21
2
2
1
3
2,088 15
31
4
2
3
1 2
3
1 3,456 3
2
3
1 2
3
1 2
3
1 8,478 13
2
2
1
3
10,152 16
2
2
1
3
2
2
1
3
2,160 3
8 53 2
3
1
2
2
3
1
2
144 1
2
2
1
4
288 1
4
2
2
4
2
4
216 2
2
3
1
2
528 3
2
2
1
4
2,016 3
21
6
8,640 1
1
8
40,320 1
2
3
1
2
2
3
1
2
792 4
2
2
1
4
1,440 3
21
6
1,440 1
2
2
1
4
2
2
1
4
576 1
521 321
3
2
3
1
2
72 1
2
3
1
2
2
3
1
2
432 2
2
2
1
4
576 1
431 32
2
1 32
2
1 24 4
321
3
72 6
31
5
240 1
2
4
192 4
2
3
1
2
396 17
2
2
1
4
768 8
21
6
720 1
321
3
321
3
108 2
2
4
720 5
2
3
1
2
720 10
2
2
1
4
288 1
31
5
2
4
2,880 1
2
3
1
2
720 1
2
4
2
4
864 2
2
3
1
2
2,592 10
2
2
1
4
7,488 7
m z
R
z
C
z
S

eg
MC
8 431 2
4
21
6
17,280 2
2
3
1
2
2
3
1
2
3,744 15
2
2
1
4
3,456 7
42
2
3
2
1
2
3
2
1
2
8 1
32
2
1 16 1
321
3
48 1
2
4
192 2
2
3
1
2
336 4
2
2
1
4
576 3
32
2
1 32
2
1 72 8
321
3
240 10
31
5
720 2
2
4
384 4
2
3
1
2
1,104 23
2
2
1
4
2,880 15
21
6
5,760 2
321
3
321
3
360 4
2
4
1,728 6
2
3
1
2
2,448 17
2
2
1
4
1,728 3
31
5
2
4
5,760 1
2
3
1
2
2,880 1
2
4
2
4
1,296 4
2
3
1
2
5,184 11
2
2
1
4
19,584 12
21
6
69,120 3
1
8
241,920 1
2
3
1
2
2
3
1
2
10,368 24
2
2
1
4
15,552 15
21
6
8,640 1
2
2
1
4
2
2
1
4
3,456 2
3
2
2 3
2
2 3
2
2 4 1
3
2
1
2
8 1
32
2
1 48 4
321
3
144 4
31
5
480 1
2
4
192 3
2
3
1
2
720 11
2
2
1
4
2,640 11
21
6
10,080 3
1
8
40,320 1
3
2
1
2
3
2
1
2
16 1
32
2
1 104 7
321
3
240 5
31
5
480 1
2
4
480 4
2
3
1
2
1,032 14
2
2
1
4
1,920 7
21
6
1,440 1
32
2
1 32
2
1 396 29
321
3
1,020 43
31
5
2,640 6
2
4
1,440 15
2
3
1
2
4,008 84
2
2
1
4
9,792 51
21
6
18,720 7
321
3
321
3
1,440 12
31
5
720 1
2
4
4,032 14
2
3
1
2
6,336 44
2
2
1
4
5,184 9
m z
R
z
C
z
S

eg
MC
8 3
2
2 31
5
2
4
11,520 2
2
3
1
2
7,200 3
2
4
2
4
4,896 8
2
3
1
2
14,832 31
2
2
1
4
46,080 25
21
6
146,880 6
1
8
483,840 1
2
3
1
2
2
3
1
2
26,208 53
2
2
1
4
6,912 2
21
6
17,280 2
2
2
1
4
2
2
1
4
6,912 2
51
3
321
3
2
3
1
2
216 1
2
3
1
2
2
3
1
2
864 1
421
2
421
2
32
2
1 16 2
2
4
144 2
321
3
24 1
2
3
1
2
192 4
2
2
1
4
96 1
3
2
1
2
3
2
1
2
16 1
32
2
1 48 3
2
4
384 3
321
3
96 2
2
3
1
2
432 5
2
2
1
4
192 1
32
2
1 32
2
1 240 19
2
4
960 10
41
4
96 1
321
3
528 22
2
3
1
2
1,968 41
31
5
480 1
2
2
1
4
2,112 11
2
4
2
4
2,592 4
41
4
576 1
321
3
3,168 12
2
3
1
2
8,208 16
31
5
5,760 2
2
2
1
4
15,552 9
21
6
8,640 1
41
4
2
3
1
2
288 1
321
3
321
3
288 3
2
3
1
2
2,160 17
2
3
1
2
2
3
1
2
9,648 21
2
2
1
4
3,168 4
3
2
1
2
3
2
1
2
3
2
1
2
32 1
32
2
1 192 4
2
4
1,248 5
41
4
96 1
321
3
288 2
2
3
1
2
1,248 7
2
2
1
4
576 2
32
2
1 32
2
1 800 28
2
4
3,648 19
41
4
192 1
321
3
1,344 240
2
3
1
2
5,184 55
31
5
960 1
2
2
1
4
4,608 12
2
4
2
4
13,248 8
41
4
1,152 1
321
3
8,064 14
2
3
1
2
24,480 28
m z
R
z
C
z
S

eg
MC
8 3
2
1
2
2
4
31
5
11,520 1
2
2
1
4
38,016 14
21
6
17,280 1
41
4
2
3
1
2
576 1
321
3
321
3
576 3
2
3
1
2
4,176 15
2
3
1
2
2
3
1
2
19,296 23
2
2
1
4
5,184 5
32
2
1 32
2
1 32
2
1 2,768 69
2
4
9,504 59
41
4
720 6
321
3
5,328 117
2
3
1
2
18,144 206
31
5
8,640 11
2
2
1
4
26,016 77
21
6
15,840 5
2
4
2
4
27,072 16
41
4
2,304 2
321
3
22,176 77
2
3
1
2
62,784 110
31
5
48,960 9
2
2
1
4
130,176 57
21
6
207,360 7
41
4
321
3
432 2
2
3
1
2
2,880 5
321
3
321
3
4,078 31
2
3
1
2
19,512 137
2
2
1
4
4,896 9
2
3
1
2
2
3
1
2
72,576 133
31
5
8,640 4
2
2
1
4
47,232 42
2
4
2
4
2
4
67,824 8
41
4
5,184 2
321
3
69,120 14
2
3
1
2
177,120 25
31
5
172,800 3
2
2
1
4
475,200 20
21
6
1,296,000 5
1
8
3,628,800 2
41
4
41
4
576 1
321
3
3,456 2
2
3
1
2
12,096 3
2
2
1
4
3,456 1
321
3
321
3
27,216 22
2
3
1
2
90,720 54
31
5
8,640 1
2
2
1
4
58,752 10
2
3
1
2
2
3
1
2
263,952 53
31
5
86,400 3
2
2
1
4
302,400 30
21
6
129,600 2
2
2
1
4
2
2
1
4
51,840 4
41
4
2
3
1
2
2
3
1
2
4,320 2
321
3
321
3
2
3
1
2
4,752 10
2
3
1
2
2
3
1
2
36,288 24
2
3
1
2
2
3
1
2
2
3
1
2
167,184 27
2
2
1
4
33,696 7
Ma h. Me h. Appl. Si.
2017, 00 1{25 Copy igh


2017 John Wiley & Sons, L d.
17
P epa ed using mmaau h.ls
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
Sho ly a e , Lyakh [38℄ de e mined 21 ongu a ions wi h poin ank 8, whih an be iden ied wi h he pa ial La in squa es
1 2
3 4
1 2
3 4
1234
3 4
1 2
1 2 3 4
2 1 4 3
1234
4 3
2 1
1 2 3
2 1 4
3 4
1 2 3 4
2 4
1 3
1234
1 3
4 2
F
1
F
2
F
3
F
4
F
5
F
6
F
7
2 4
4 1
2 3
1 3
2 4
4 1
2 3
3 1
2 4
1 3
4 3
2 1
2 4
3 1
4 3
2 1
3 2 4
1 3 2
4 1
4 1 3 2
2 3 4 1
3 4 2
1 2 3
4 1
F
8
F
9
F
10
F
11
F
12
F
13
F
14
132
321
2 1
423
321
1 4
243
2 1 3
1 4
234
132
4 1
3 4 2
123
4 1
342
2 1 3
4 1
432
321
4 1
F
15
F
16
F
17
F
18
F
19
F
20
F
21
They o espond in Table 7 o
i. The wo main lasses o ype (4
2
;
2
4
;
2
4
):
F
3
and
F
13
.
ii. The ou main lasses o ype (42
2
;
2
4
;
2
4
):
F
2
,
F
4
,
F
6
and
F
7
.
iii. The main lass o ype (3
2
2
;
3
2
2
;
3
2
2):
F
15
.
i . The h ee main lasses o ype (3
2
2
;
3
2
2
;
2
4
):
F
5
,
F
12
and
F
14
.
. The six main lasses o ype (3
2
2
;
2
4
;
2
4
): om
F
16
o
F
21
.
i. Fi e o he eigh main lasses o ype (2
4
;
2
4
;
2
4
):
F
1
,
F
8
,
F
9
,
F
10
and
F
11
.
The nex wo main lasses o ype (2
4
;
2
4
;
2
4
) omple e he lis o Lyakh.
1 2
2 1
3 4
4 3
1 2
3 4
4 2
3 1
F
22
F
23
The eigh h main lass o ype (2
4
;
2
4
;
2
4
) is no ela ed o a ongu a ion beause he e exis non-onne ed poin s in he
o esponding semine (see Figu e 8).

1 2
2 1
3 4
4 3
Figu e 8. Semine o poin ank 8 ha is no a ongu a ion.
18 Copy igh


2017 John Wiley & Sons, L d.
Ma h. Me h. Appl. Si.
2017
,001{25
P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
6. Bina y ons ain s ela ed o he se s
TS
n
and
TCO
n
This se ion deals wi h a se ies o bina y ons ain s ha ha a e ize he se s o o ally symme i and o ally onjuga e
o hogonal pa ial La in squa es o gi en o de and weigh . He ea e , in o de o a oid degene ay, pa ial La in squa es a e
assumed o ha e a leas one en y in eah ow, a leas one en y in eah olumn, and a leas one opy o eah symbol. F om
Theo em 2.1, he ollowing sys em o ons ain s mus , he e o e, hold.

































x
ijk
x
i
0
j k
= 0
;
o all
i ; i
0
; j ; k

n
suh ha
i
6
=
i
0
;
x
ijk
x
i j
0
k
= 0
;
o all
i ; j ; j
0
; k

n
suh ha
j
6
=
j
0
;
x
ijk
x
ijk
0
= 0
;
o all
i ; j ; k ; k
0

n
suh ha
k
6
=
k
0
;
P
j ;k
2
[
n
℄
x
ijk

1
;
o all
i
2
[
n
℄
;
P
i ;k
2
[
n
℄
x
ijk

1
;
o all
j
2
[
n
℄
;
P
i ;j
2
[
n
℄
x
ijk

1
;
o all
k
2
[
n
℄
;
x
ijk
2
0
;
1
g
;
o all
i ; j ; k

n :
(2)
Lemma 6.1
Le
n
and
m
be wo posi i e in ege s suh ha
n

m

n
2
.
a) I
m > n
, hen e e y pai o o hogonal onjuga es o a pa ial La in squa e in he se
TCO
n
;
m
a e dis in .
b) I
j
TCO
n
;
m
j
= 0
, hen
j
TCO
n
;
m
0
j
= 0
, o all
m
0
2
m
+ 1
;:::;n
2
g
.
P o o .
Le us p o e eah s a emen sepa a ely.
a) Le
P
2 R
n;n ;n
;
m
and
 ; 
0
2
S
3
be suh ha

6
=

0
and
P

=
P

0
. Sine
m > n
, he e exis s one symbol
k
2
[
n
℄ and a
dis in pai o elemen s (
i
1
; j
1
) and, (
i
2
; j
2
) in [
n
℄

[
n
℄ suh ha
(
i
1
; j
1
; k
)
;
(
i
2
; j
2
; k
)
g 
E
(
P

)
E
(
P

0
). As a onsequene,
P

=
P

0
is no o hogonal o i sel .
b) O he wise, he pa ial La in squa e ha esul s a e emp ying any
m
0

m
lled ells o he pa ial La in squa e in TCO
n
;
m
0
would be in TCO
n
;
m
, whih is a on adi ion.
Lemma 6.1.a does no hold in gene al in ase o being
m
=
n
. Thus, o ins ane, he pa ial La in squa e
P
2 R
3
;
3
;
3;3
suh
ha
E
(
P
) =
(1
;
1
;
1)
;
(2
;
2
;
2)
;
(3
;
3
;
3)
g
is o ally symme i and o hogonal o i sel .
Based on (2), we es ablish in Se ion 3 some equa ions o deal, espe i ely, wi h he se s TS
n
and TCO
n
. To his end, le
us in odue he ollowing no a ion
x

i
1
i
2
i
3
:=
x
i

(1)
i

(2)
i

(3)
;
o all

2
S
3
and
x
i
1
i
2
i
3
2
X
g
. Besides, we label he six pe mu a ions in
S
3
as
S
3
:=

1
= Id
; 
2
= (12)
; 
3
= (13)
; 
4
= (23)
; 
5
= (123)
; 
6
= (132)
g
:
P op osi ion 6.2
Le
n
and
m
be wo posi i e in ege s suh ha
n < m

n
2
. Then,
a) The se
TS
n
is iden ied wi h he se o ze os o (2) and
x

s
ijk
=
x
ijk
;
o all
i ; j ; k
2
[
n
℄
and
s
2
1
;
2
;
3
g
:
(3)
b) The se
TS
n
;
m
is iden ied wi h he se o ze os o (2){(3) and
X
i ;j ;k
2
[
n
℄
x
ijk
=
m :
(4)
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P epa ed using mmaau h.ls
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
) The se
TCO
n
is iden ied wi h he se o ze os o (2) and
x

s
ijp
x

s
k l p
x

ijq
x

k l q
= 0
;
o all
i ; j ; k ; l ; p; q

n
;
s ;

3;
suh ha
(
i ; j
)
6
= (
k ; l
)
; s

:
(5)
d) The se
TCO
n
;
m
is iden ied wi h he se o ze os o (2), (4) and (5).
P o o .
The esul ollows s aigh o wa dly om he deni ions exposed in Se ion 2 one eah pa ial La in squa e
P
= (
p
i j
)
2
R
;s ;n
is iden ied wi h a ze o (
x
111
;:::; x
s n
) suh ha
x
ijk
= 1 i
p
i j
=
k
and 0, o he wise. Thus, o ins ane, i we ous
on he p oo o s a emen (), hen, gi en 1

s <

3, he sys em o equa ions de e mined by (5) in ol es he


1
s
- and


1
-onjuga es o
P
o be o hogonal. Besides, om Lemma 6.1.a, bo h onjuga es a e dis in .
P oposi ion 6.2 has been implemen ed in he CSP sol e Minion [68℄ o ob ain he nume ial da a exposed in Table 8. Fu he ,
Table 9india es he un ime ha is equi ed in ou ompu e sys em (
In el Co e i7-2600, wi h a 3.4 GHz p oesso and 16
GB o RAM
) o de e mine one spei example in he se s TS
n
;
m
and TCO
n
;
m
.
m
j
TS(
n
;
m
)
j j
TCO(
n
;
m
)
j
n n
3 4 5 6 3 4
3 1 36
4 6 1 216 576
5 6 12 1 12 45168
6 10 24 20 1 0 315048
7 12 64 80 30 0 391824
8 3 60 220 210 0 95028
9 3 100 380 680 0 2616
10 148 910 1980 0
11 72 1010 4380 0
12 90 1630 7660 0
13 72 2740 17820 0
14 36 2040 23370 0
15 16 2784 37476 0
16 16 3395 68850 0
17 2195 68190
18 2080 96660
19 2320 145560
20 900 122040
21 900 146040
22 480 196200
23 240 132480
24 30 148710
25 30 157320
26 101430
27 81540
28 86310
29 35820
30 33390
31 20340
32 11340
33 4560
34 3960
35 720
36 480
To al 41 711 24385 1755547 264 850260
Table 8.
Dis ibu ion o he se s TS
n
;
m
and TCO
n
;
m
.
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P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
Run ime (seonds) Run ime (seonds)
n m
TS
n
;
m
TCO
n
;
m
5 5
<
1 22
10
<
1 3
6 6
<
1 8561
12
<
1 10
15
<
1 74
10 10 69 Ou o memo y
50
<
1 "
15 15
>
3 hou s "
60 2 "
20 100 Ou o memo y "
Table 9.
Run imes equi ed o ge exa ly one o ally symme i o o ally onjuga e o hogonal pa ial La in squa e o a gi en
o de and weigh .
7. Lie pa ial quasig oup ings de i ed om he onjuga e-ex ension o a pa ial La in
squa e
The inlusion o new bina y ons ain s in o (2){(5) enables us o de e mine amilies o pa ial La in squa es in he se s TS
n
and
TCO
n
wi h possible applia ions in dis in elds. As an illus a i e example, we onlude his pape by des ibing in his se ion
a new amily o Lie pa ial quasig oup ings ela ed o a o ally symme i pa ial La in squa e o o de 3
n
, whih is de i ed in
u n om a gi en pa ial La in squa e o o de
n
. Reall ha a
Lie algeb a
is an an i-ommu a i e algeb a
A
ha holds he
so-alled
Jaobi iden i y
J
(
a; b ; 
) := (
ab
)

+ (
b 
)
a
+ (
 a
)
b
= 0
;
o all
a; b ; 
2
A:
(6)
Le
P
= (
p
i j
)
2 R
n;n ;n
;
m
. We dene he
n

n
a ays
P
0
= (
p
0
i j
) and
P
00
= (
p
00
i j
) suh ha
p
0
i j
:=



p
i j
+
n ;
i
p
i j
2
[
n
℄
;
0
;
o he wise
:
and
p
00
i j
:=



p
i j
+ 2
n ;
i
p
i j
2
[
n
℄
;
0
;
o he wise
:
(7)
Then, we dene he pa ial La in squa e
P
= (
p
i j
)
2 R
3
n;
3
n;
3
n
;6
m
by means o nine
n

n
bloks as
P
:

0
P
00
P
0
(23)
P
00
(12)
0
P
(132)
P
0
(123)
P
(13)
0
(8)
whe e
0
deno es he
n

n
a ay wi h all i s en ies being ze o. We all his new pa ial La in squa e he
onjuga e-ex ension
o
P
. Thus, o ins ane, Figu e 9shows he onjuga e-ex ension o he pa ial La in squa e exposed in Figu e 2.
7 8 4 5
9 5
7 6
7 1
8 9 1 2
7 3
4 6 1 3
5 1
5 2
Figu e 9. Conjuga e-ex ension o he pa ial La in squa e
P
2 R
3
;
3
;
3
o Figu e 2.
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P epa ed using mmaau h.ls

Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
Lemma 7.1
I
P
2 R
n;n ;n
;
m
, hen
P
2
TS
3
n
;6
m
.
P o o .
The esul ollows om he en y se
E
(
P
) one we keep in mind (7) and (8).
Le
A
K
(
P
) deno e he pa ial quasig oup ing o e a ni e eld
K
o ha a e is i wo ha is ela ed o
P
. Pa iula ly, we
ous on he ase o being
P
2
TS
n
. I his is he ase, hen he deni ion (8) o he pa ial La in squa e
P
esul s
P

0
P
00
P
0
P
00
0
P
P
0
P
0
(9)
Theo em 7.2
Le
K
be a ni e eld o ha a e is i wo and le
P
2
TS
n
be he mul iplia ion able o a quasig oup
([
n
℄
;

)
sa is ying he
le in e i e law
(
a

b
)


= (


b
)

a;
o all
a; b ; 
2
[
n
℄
:
(10)
Then, he pa ial quasig oup ing
A
K
(
P
)
is a Lie algeb a.
P o o .
The symme y o he pa ial La in squa e
P
= (
p
i j
), wi h
p
i i
= 0, o all
i

3
n
, oge he wi h he a o being
K
a
ni e eld o ha a e is i wo, in ol es
A
K
(
P
) o be an i-ommu a i e. Now, in o de o p o e ha he Jaobi iden i y (6)
holds, suppose
e
1
;:::;e
3
n
g
o be he basis o
A
K
(
P
), whih we pa i ion in o he h ee se s
e
1
;:::;e
n
g
,
e
n
+1
;:::;e
2
n
g
and
e
2
n
+1
;:::;e
3
n
g
. Le
S
(
e
i
) deno e whih one o hese h ee se s on ains eah basis e o
e
i
. F om (9), we ha e
ha , i
S
(
e
i
) =
S
(
e
j
), hen
e
i
e
j
= 0. Besides, i
S
(
e
i
)
6
=
S
(
e
j
) and
e
i
e
j
6
= 0, hen
S
(
e
i
)
6
=
S
(
e
i
e
j
)
6
=
S
(
e
j
). As a onsequene,
J
(
e
i
; e
j
; e
k
) = 0, o all
i ; j ; k

3
n
suh ha he h ee se s
S
(
e
i
),
S
(
e
j
) and
S
(
e
k
) ei he oinide o a e pai wise dis in .
Then, om he symme y o he Jaobi iden i y, i is enough o ous on he exp ession
J
(
e
i
; e
j
; e
k
) in ase o being
S
(
e
i
) =
S
(
e
j
)
6
=
S
(
e
k
). I his is he ase,
e
i
e
j
= 0 and hene,
J
(
e
i
; e
j
; e
k
) = (
e
j
e
k
)
e
i
+ (
e
k
e
i
)
e
j
=
e
(
j

k
)

i
+
e
(
k

i
)

j
. The esul
ollows om he symme y o he pa ial La in squa e
P
and he le in e i e law.
E e y o ally symme i pa ial La in squa e sa is ying (10 ) ons i u es he mul iplia ion able o a pa ial o ally symme i
g oup. In o de o ompu e his kind o pa ial La in squa es, we inlude he ollowing equa ions o (2){(4)
x
ijk
x
k l s
x
lj
(
x
i s

1) = 0
;
o all
i ; j ; k ; l ; s ;
2
[
n
℄ (11)
X
k

n
x
ijk

1
! X
k

n
x
ljk
!
x
lj
X
k

n
x
i k
!
= 0
;
o all
i ; j ; l ;
2
[
n
℄ (12)
x
ijk
X
s

n
x
k l s

1
! X
s

n
x
ljs
!
x
lj
X
s

n
x
i s
!
= 0
;
o all
i ; j ; k ; l ;
2
[
n
℄ (13)
The implemen a ion o hese equa ions in o ou CSP sol e de e mines, o ins ane, he pai o pa ial La in squa es exposed
in Figu e 10 , whih gi e ise in u n, ao ding o Theo em 7.2, o a pai o Lie pa ial quasig oup ings as we ha e p e iously
des ibed.
3 1
2
1 3
2 1
1 2
4 3
3 4
6 5
5 6
Figu e 10. To ally symme i pa ial La in squa es sa is ying he le in e i e law.
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P epa ed using mmaau h.ls
R. M. Falon, O. J. Falon, J. Nu~nez
Ma hema ial
Me hods in he
Applied Sienes
8. Conlusion and u he s udies
This pape has deal wi h he enume a ion and lassia ion o pa ial La in e angles and semine s by means o ompu a ional
algeb ai geome y. Bo h ombina o ial s u u es ha e been iden ied wi h he poin s o aÆne a ie ies dened by ze o-
dimensional adial ideals o polynomials. Thei deomposi ions in o ni ely many disjoin subse s, eah o hem being he ze os
o a iangula sys em o polynomial equa ions, ha e eme ged as a use ul ehnique o de e mine, by means o he ompu e
algeb a sys em Singula , he dis ibu ion o

s
pa ial La in e angles based on [
n
℄ in o iso opi and main lasses ao ding
o hei weigh and ypes, o all
; s ; n

6, and ha o non-omp essible egula pa ial La in squa es o o de
n

8. The
la e is equi alen o ha o semine s wi h poin ank up o eigh and has enabled us o omple e a lassia ion p e iously
es ablished by Lyakh [38℄. Gene al o mulas o he numbe o pa ial La in squa es o weigh up o six and a ensus o all he
semine s wi h a mos six poin s ha e also been es ablished. A on enien gene aliza ion o he ompu a ional me hod exposed
in his pape o he heo y o
k
-semine s and ha o non-omp essible, egula and mu ually egula ly o hogonal pa ial La in
squa es de eloped by Usan [12℄ is es ablished as u he wo k. We ha e also des ibed a se ies o bina y ons ain s ha enable
us o de e mine he dis ibu ion o he se s TS
n
and TCO
n
o o ally symme i and o ally onjuga e pa ial La in squa es o
o de
n
, espe i ely, ao ding o hei weigh s. By means o he CSP sol e Minion, we ha e ompu ed he o me , o all
2

n

6, and he la e , o all 2

n

4. A u he s udy o imp o e he eÆieny o he p oposed me hod is equi ed o
deal wi h highe o de s. Besides, we ha e in odued he onjuga e-ex ension o a gi en pa ial La in squa e, whih gi es ise o
a o ally symme i pa ial La in squa e. Pa iula ly, he des ip ion o a amily o Lie pa ial quasig oup ings de i ed om he
onjuga e-ex ension o a o ally symme i pa ial La in squa e ha holds he le in e i e law has enabled us o del e in o he
open p oblem o ons u ing examples o his ype o Lie algeb as.
Re e enes
1. Hulpke A, Kaski P,

Os e ga d PRJ. The numbe o La in squa es o o de 11.
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2011; 80: 1197{1219.
DOI: 10.1090/S0025-5718-2010-02420-2.
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
8 La in squa es.
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1990; 54: 143{148.
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2007; 15: 98{119.
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Annals o Combina o is
2005; 9: 335{344. DOI: 10.1007/s00026-005-
0261-7.
5. S ones DS. The many o mulae o he numbe o La in e angles.
Ele oni Jou nal o Combina o is
2010; 17 1, 46 pp.
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G aphs and Combina o is
2016; 32: 1187-1202.
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Dis e e Ma hema is
2013; 313: 1150{1161. DOI:
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8. Falon RM. Enume a ion and lassia ion o sel -o hogonal pa ial La in e angles by using he polynomial me hod.
Eu opean
Jou nal o Combina o is
2015; 48: 215{223. DOI: 10.1016/j.ej.2015.02.022.
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Ele oni No es in Dis e e Ma hema is
2015; 49: 765{771. DOI:
10.1016/j.endm.2015.06.103.
10. Baye D.
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11. Falon RM, Ma n-Mo ales J. G obne bases and he numbe o La in squa es ela ed o au o opisms o o de up o 7.
Jou nal o
Symboli Compu a ion
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Ma ema iki Bil en
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means o a SAT sol e . In: Vigo-Aguia , J.
P oeedings o 17 h In e na ional Con e ene Compu a ional and Ma hema ial Me hods
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2017, 00 1{25 Copy igh


2017 John Wiley & Sons, L d.
23
P epa ed using mmaau h.ls
Ma hema ial
Me hods in he
Applied Sienes R. M. Falon, O. J. Falon, J. Nu~nez
16. Bailey RA. Enume a ion o o ally symme i La in squa es.
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Co igendum
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