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On Computing the Number of Latin Rectangles

2015
*
Graphs and Combinatorics
*

Doyle (circa 1980) found a formula for

doi:10.1007/s00373-015-1643-1
fatcat:w4ngciaofzdidp2ouapvrbecxq
*the**number**of*k ×n*Latin**rectangles*L k,n . This formula remained dormant until it was recently used for counting k × n*Latin**rectangles*, where k ∈ {4, 5, 6}. ... Motivated by*computational*data for 3 ≤ k ≤ 6, some research problems and conjectures about*the*divisors*of*L k,n are presented. ... Open Access This article is distributed under*the*terms*of**the*Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution ...##
###
Latin Squares of Order 10

1995
*
Electronic Journal of Combinatorics
*

We describe two independent

doi:10.37236/1222
fatcat:iwwlpqtzk5d23ha2d2qmzzqduy
*computations**of**the**number**of**Latin*squares*of*order 10. ... We also give counts*of**Latin**rectangles*with up to 10 columns, and estimates*of**the**number**of**Latin*squares*of*orders up to 15. ... Clearly,*the*rows*of*a*Latin**rectangle*R correspond to*the**one*-factors in a*one*-factorization*of*G(R). ...##
###
The number of latin squares of order eight

1967
*
Journal of Combinatorial Theory
*

When k = K (Sade used K = 4),

doi:10.1016/s0021-9800(67)80021-8
fatcat:hfpanykyvbeojoup7i6x32fx5q
*one*may sum*the*product*of**the**number**of*ways in which a*rectangle*could have been formed (already known) and*the**number**of*ways*the**rectangle*can be completed to a square ... (easily*computed*) over*the*inequivalent K-row*rectangles*, producing*the**number**of*reduced n x n squares. ...##
###
Computing autotopism groups of partial Latin rectangles: a pilot study
[article]

2019
*
arXiv
*
pre-print

*Computing*

*the*autotopism group

*of*a partial

*Latin*

*rectangle*can be performed in a variety

*of*ways. ...

*Latin*squares is significantly poorer than other partial

*Latin*

*rectangles*

*of*comparable size, obstructed by

*the*existence

*of*

*Latin*squares with large (possibly transitive) autotopism groups. ... Falcón's work is partially supported by

*the*research project FQM-016 from Junta de Andalucía, and

*the*Departmental Research Budget

*of*

*the*Department

*of*Applied Mathematics I

*of*

*the*University

*of*Seville ...

##
###
Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs
[article]

2022
*
arXiv
*
pre-print

For small parameter values where no Youden

arXiv:1910.02791v3
fatcat:p4ojrmoszzbphaozvspaptodla
*rectangles*exist, we also enumerate*rectangles*where*the**number**of*symbols common to two columns is always*one**of*two possible values, differing by 1, which we ... We have enumerated all Youden*rectangles*for a range*of*small parameter values, excluding*the*almost square cases where k = n-1, in a large scale*computer*search. ... Acknowledgments*The**computational*work was performed*on*resources provided by*the*Swedish National Infrastructure for*Computing*(SNIC) at High Performance*Computing*Center North (HPC2N). ...##
###
The number of Latin rectangles
[article]

2007
*
arXiv
*
pre-print

We show how to generate an expression for

arXiv:math/0703896v1
fatcat:w26zzme7dfaeldtl2yedhv63ka
*the**number**of*k-line*Latin**rectangles*for any k. ...*The**computational*complexity*of**the*resulting expression, as measured by*the**number**of*additions and multiplications required to evaluate it, is*on**the*order*of*n^(2^(k-1)). ... When we talk about "*the**number**of*k-line*Latin**rectangles*", we really mean*the*function L k . ...##
###
Triples of Orthogonal Latin and Youden Rectangles For Small Orders
[article]

2018
*
arXiv
*
pre-print

We have performed a complete enumeration

arXiv:1810.12639v1
fatcat:x7dfexkfjrgl5pimfyw7s56mbi
*of*non-isotopic triples*of*mutually orthogonal k× n*Latin**rectangles*for k≤ n ≤ 7. ... We have also studied orthogonal triples*of*k × 8*rectangles*which are formed by extending mutually orthogonal triples with non-trivial autotopisms*one*row at a time, and requiring that*the*autotopism group ... Acknowledgments*The**computational*work was performed*on*resources provided by*the*Swedish National Infrastructure for*Computing*(SNIC) at High Performance*Computing*Center North (HPC2N) . ...##
###
Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

2018
*
Mathematical methods in the applied sciences
*

As a by-product, explicit formulas are determined for

doi:10.1002/mma.4820
fatcat:7gaagajuzbf3hkhmwcg6j6yvoe
*the**number**of*partial*Latin**rectangles**of*size up to six. ... This paper provides an in-depth analysis*of*how*computational*algebraic geometry can be used to deal with*the*problem*of*counting and classifying r× s partial*Latin**rectangles*based*on*n symbols*of*a given ...*The*rest*of*components*of*T ′ are zeros and do not have any influence*on**the**number**of*partial*Latin**rectangles*having T ′ as row, column or symbol type. ...##
###
In Search of Balance: The Challenge of Generating Balanced Latin Rectangles
[chapter]

2017
*
Lecture Notes in Computer Science
*

In this work, we study some

doi:10.1007/978-3-319-59776-8_6
fatcat:gpi5wupbhnhubgjivzfqt2q7vm
*of**the*properties*of*balanced*Latin**rectangles*, prove*the*nonexistence*of*perfect balance for an infinite family*of*sizes, and present several methods to generate*the*most balanced ... Balanced*Latin**Rectangles*appear to be even more defiant than balanced*Latin*Squares, to such an extent that perfect balance may not be feasible for*Latin**rectangles*. ... This work was supported by*the*National Science Foundation (NSF Expeditions in*Computing*awards for*Computational*Sustainability, grants CCF-1522054 and CNS-0832782, NSF*Computing*research infrastructure ...##
###
Enumerating Partial Latin Rectangles

2020
*
Electronic Journal of Combinatorics
*

This paper deals with different

doi:10.37236/9093
fatcat:nn2iywwscvhaplc7mqoujryo4i
*computational*methods to enumerate*the*set $\mathrm{PLR}(r,s,n;m)$*of*$r \times s$ partial*Latin**rectangles**on*$n$ symbols with $m$ non-empty cells. ... Adapting Sade's method for enumerating*Latin*squares, we*compute**the*exact size*of*$\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n= ... Thanks also to Daniel Kotlar for assistance in*computing*autotopism groups, which is leading to*the*papers [17, 18, 69] . ...##
###
An IoT Scheduling and Interference Mitigation Scheme in TSCH Using Latin Rectangles

2019
*
2019 IEEE Global Communications Conference (GLOBECOM)
*

Time Slotted Channel Hopping (TSCH) is

doi:10.1109/globecom38437.2019.9013482
dblp:conf/globecom/BoucettaNML19
fatcat:utpvrn5e4vh6phlwzsgrpcz6bm
*one**of**the*most used MAC mechanisms introduced by*the*new amendment IEEE 802.15.4e. ... In essence,*the*scheduling*of*links is performed by*Latin**rectangles*where rows are channel offsets and columns are slot offsets. ... Thus, our approach fits into that principle. 2) TSCH Frame*Computation*:*The*size*of**the**Latin**rectangle*is defined by*the**number**of*channels and*the**number**of*time slots. ...##
###
Enumerating partial Latin rectangles
[article]

2019
*
arXiv
*
pre-print

This paper deals with distinct

arXiv:1908.10610v1
fatcat:hkvg24wbknci3mvcuhfbavqtiq
*computational*methods to enumerate*the*set PLR(r,s,n;m)*of*r × s partial*Latin**rectangles**on*n symbols with m non-empty cells. ... Adapting Sade's method for enumerating*Latin*squares, we*compute**the*exact size*of*PLR(r,s,n;m), for all r ≤ s ≤ n ≤ 7, and all r ≤ s ≤ 6 when n=8. ... Stones would also like to acknowledge*the*use*of*math.stackexchange.com for discussing problems arising in this work. ...##
###
Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

2015
*
European journal of combinatorics (Print)
*

*The*current paper deals with

*the*enumeration and classification

*of*

*the*set SOR_r,n

*of*self-orthogonal r× r partial

*Latin*

*rectangles*based

*on*n symbols. ...

*The*distribution

*of*r× s partial

*Latin*

*rectangles*based

*on*n symbols according to their size is also obtained, for all r,s,n≤ 4. ... Its

*number*

*of*filled cells is its size. Let R r,s,n and R r,s,n:m respectively denote

*the*set

*of*r × s partial

*Latin*

*rectangles*based

*on*[n] and its subset

*of*partial

*Latin*

*rectangles*

*of*size m. ...

##
###
Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method
[chapter]

2013
*
The Seventh European Conference on Combinatorics, Graph Theory and Applications
*

*The*current paper deals with

*the*enumeration and classification

*of*

*the*set SOR r,n

*of*self-orthogonal r × r partial

*Latin*

*rectangles*based

*on*n symbols. ...

*The*distribution

*of*r × s partial

*Latin*

*rectangles*based

*on*n symbols according to their size is also obtained, for all r, s, n ≤ 4. ... Its

*number*

*of*filled cells is its size. Let R r,s,n and R r,s,n:m respectively denote

*the*set

*of*r × s partial

*Latin*

*rectangles*based

*on*[n] and its subset

*of*partial

*Latin*

*rectangles*

*of*size m. ...

##
###
Automatic Counting of Generalized Latin Rectangles and Trapezoids
[article]

2021
*
arXiv
*
pre-print

many terms

arXiv:2108.11285v1
fatcat:bdxbbeoygbc4ticcody54lkt4a
*of*'hard to*compute*sequences', namely*the**number**of**Latin*trapezoids, generalized derangements, and generalized three-rowed*Latin**rectangles*. ... At*the*end we also sketch*the*proof*of*a generalization*of*Ira Gessel's 1987 theorem that says that for any*number**of*rows, k,*the**number**of**Latin**rectangles*with k rows and n columns is P-recursive in ... A k × n*Latin**rectangle*is a k × n array*of*integers where every*one**of**the*k rows is a permutation*of*{1, 2, . . . , n}, and*the*entries*of*each column are distinct. ...
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