On Computing the Number of Latin Rectangles

@article{Stones2016OnCT,
  title={On Computing the Number of Latin Rectangles},
  author={Rebecca J. Stones and Sheng Lin and X. Liu and G. Wang},
  journal={Graphs and Combinatorics},
  year={2016},
  volume={32},
  pages={1187-1202}
}
Doyle (circa 1980) found a formula for the number of $$k \times n$$k×n Latin rectangles $$L_{k,n}$$Lk,n. This formula remained dormant until it was recently used for counting $$k \times n$$k×n Latin rectangles, where $$k \in \{4,5,6\}$$k∈{4,5,6}. We give a formal proof of Doyle’s formula for arbitrary k. We also improve a previous implementation of this formula, which we use to find $$L_{k,n}$$Lk,n when $$k=4$$k=4 and $$n \le 150$$n≤150, when $$k=5$$k=5 and $$n \le 40$$n≤40 and when $$k=6$$k=6… 

Enumerating Partial Latin Rectangles

TLDR
It is proved that the size of the set of partial Latin rectangles on symbols with non-empty cells is a symmetric polynomial of degree 3m, and the leading terms are determined using inclusion-exclusion.

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

This paper provides an in‐depth analysis of how computer algebra systems and CSP solvers can be used to deal with the problem of enumerating and distributing the set of r×s partial Latin rectangles

Latin Squares and Their Applications to Cryptography

A latin square of order-n is an n × n array over a set of n symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic

Comments on "Discrete Groups, Expanding Graphs and Invariant Measures", by Alexander Lubotzky

This document is a collection of comments that I wrote down while reading the first four chapters of the book "Discrete Groups, Expanding Graphs and Invariant Measures" by Alexander Lubotzky. Most of

References

SHOWING 1-10 OF 41 REFERENCES

The Many Formulae for the Number of Latin Rectangles

TLDR
The method of Sade in finding $L_{7,7}$, an important milestone in the enumeration of Latin squares, but which was privately published in French, is described in detail.

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of $${\mathbb{Z}_{n}}$$ is an injective map $${\sigma : S \rightarrow \mathbb{Z}_{n}}$$ such that $${S \subseteq \mathbb{Z}_{n}}$$ and σ(i)–i ≢ σ(j)− j (mod n) for distinct

An Asymptotic Series for the Number of Three-Line Latin Rectangles

$f(n,k)\sim e^{-k(k-1)/2}(n1)^{k}$ . And for the special ease of $f(n, 3),$ $nu\varphi erous$ results are reported to be obtained by authors of the United States and other countries thongh we have

The Conjectures of Alon--Tarsi and Rota in Dimension Prime Minus One

  • D. Glynn
  • Mathematics
    SIAM J. Discret. Math.
  • 2010
TLDR
Rota's basis conjecture is true for a vector space of dimension p-1 over any field of characteristic zero or p, and all other characteristics except possibly a finite number are shown.

The Asymptotic Number of Latin Rectangles

1. Introduction. The problem of enumerating n by k Latin rectangles was solved formally by MacMahon [4] using his operational methods. For k = 3, more explicit solutions have been given in [1], [2],

Cycle structures of autotopisms of the Latin squares of order up to 11

TLDR
This paper gives a classification of all autotopisms of the Latin squares of order up to 11 and studies some properties of these cycle structures.

On the Number of Latin Squares

Abstract.We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that

Proof of the Alon-Tarsi Conjecture for n=2rp

TLDR
This note proves the extended Alon-Tarsi conjecture for prime orders p, which implies that both conjectures are true for any $n$ of the form $2^rp$ with $p$ prime.

Counting three-line Latin rectangles

A k × n Latin rectangle is a k × n array of numbers such that (i) each row is a permutation of [n] = {1, 2, . . . , n} and (ii) each column contains distinct entries. If the first row is 12 · · ·n,

Asymptotic enumeration of Latin rectangles