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On Computing the Number of Latin Rectangles

Rebecca J. Stones, Sheng Lin, Xiaoguang Liu, Gang Wang
2015 Graphs and Combinatorics  
Doyle (circa 1980) found a formula for 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  ... 
doi:10.1007/s00373-015-1643-1 fatcat:w4ngciaofzdidp2ouapvrbecxq

Latin Squares of Order 10

Brendan D. McKay, Eric Rogoyski
1995 Electronic Journal of Combinatorics  
We describe two independent 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).  ... 
doi:10.37236/1222 fatcat:iwwlpqtzk5d23ha2d2qmzzqduy

The number of latin squares of order eight

Mark B. Wells
1967 Journal of Combinatorial Theory  
When k = K (Sade used K = 4), 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.  ... 
doi:10.1016/s0021-9800(67)80021-8 fatcat:hfpanykyvbeojoup7i6x32fx5q

Computing autotopism groups of partial Latin rectangles: a pilot study [article]

Rebecca J. Stones, Raúl M. Falcón, Daniel Kotlar, Trent G. Marbach
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  ... 
arXiv:1910.10103v1 fatcat:nai6z3bulzhcfd4x3dafstipym

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

Gerold Jäger, Klas Markström, Denys Shcherbak, Lars-Daniel Öhman
2022 arXiv   pre-print
For small parameter values where no Youden 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).  ... 
arXiv:1910.02791v3 fatcat:p4ojrmoszzbphaozvspaptodla

The number of Latin rectangles [article]

Peter G. Doyle
2007 arXiv   pre-print
We show how to generate an expression for 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 .  ... 
arXiv:math/0703896v1 fatcat:w26zzme7dfaeldtl2yedhv63ka

Triples of Orthogonal Latin and Youden Rectangles For Small Orders [article]

Gerold Jäger, Klas Markström, Lars-Daniel Öhman, Denys Shcherbak
2018 arXiv   pre-print
We have performed a complete enumeration 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) .  ... 
arXiv:1810.12639v1 fatcat:x7dfexkfjrgl5pimfyw7s56mbi

Counting and enumerating partial Latin rectangles by means of computer algebra systems and CSP solvers

Raúl M. Falcón, Óscar J. Falcón, Juan Núñez
2018 Mathematical methods in the applied sciences  
As a by-product, explicit formulas are determined for 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.  ... 
doi:10.1002/mma.4820 fatcat:7gaagajuzbf3hkhmwcg6j6yvoe

In Search of Balance: The Challenge of Generating Balanced Latin Rectangles [chapter]

Mateo Díaz, Ronan Le Bras, Carla Gomes
2017 Lecture Notes in Computer Science  
In this work, we study some 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  ... 
doi:10.1007/978-3-319-59776-8_6 fatcat:gpi5wupbhnhubgjivzfqt2q7vm

Enumerating Partial Latin Rectangles

Raúl M. Falcón, Rebecca J. Stones
2020 Electronic Journal of Combinatorics  
This paper deals with different 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] .  ... 
doi:10.37236/9093 fatcat:nn2iywwscvhaplc7mqoujryo4i

An IoT Scheduling and Interference Mitigation Scheme in TSCH Using Latin Rectangles

Cherifa Boucetta, Boubakr Nour, Hassine Moungla, Laaziz Lahlou
2019 2019 IEEE Global Communications Conference (GLOBECOM)  
Time Slotted Channel Hopping (TSCH) is 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.  ... 
doi:10.1109/globecom38437.2019.9013482 dblp:conf/globecom/BoucettaNML19 fatcat:utpvrn5e4vh6phlwzsgrpcz6bm

Enumerating partial Latin rectangles [article]

Raúl M. Falcón, Rebecca J. Stones
2019 arXiv   pre-print
This paper deals with distinct 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.  ... 
arXiv:1908.10610v1 fatcat:hkvg24wbknci3mvcuhfbavqtiq

Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

Raúl M. Falcón
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.  ... 
doi:10.1016/j.ejc.2015.02.022 fatcat:63g4eo2t5rhutj7md342jf3tce

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

Raúl M. Falcón
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.  ... 
doi:10.1007/978-88-7642-475-5_96 fatcat:72rf7cbgfrauhmm6wip6jg6h7q

Automatic Counting of Generalized Latin Rectangles and Trapezoids [article]

George Spahn, Doron Zeilberger
2021 arXiv   pre-print
many terms 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.  ... 
arXiv:2108.11285v1 fatcat:bdxbbeoygbc4ticcody54lkt4a
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