Overlapping latin subsquares and full products

Browning, Joshua M., Vojtěchovský, Petr, and Wanless, Ian M.. "Overlapping latin subsquares and full products." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 175-184. <http://eudml.org/doc/37750>.

@article{Browning2010,
abstract = {We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac\{n\}\{m\}\{n\atopwithdelims ()h\}/\{m\atopwithdelims ()h\}$ subsquares of order $m$, where $h=\lceil (m+1)/2\rceil $. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt\{2m\}+2$ in $n$. (b) For all $n\ge 5$ there exists a loop of order $n$ in which every element can be obtained as a product of all $n$ elements in some order and with some bracketing.},
author = {Browning, Joshua M., Vojtěchovský, Petr, Wanless, Ian M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {latin square; latin subsquare; overlapping latin subsquares; full product in loops; Latin square; Latin subsquare; overlapping Latin subsquare; full product in loops},
language = {eng},
number = {2},
pages = {175-184},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Overlapping latin subsquares and full products},
url = {http://eudml.org/doc/37750},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Browning, Joshua M.
AU - Vojtěchovský, Petr
AU - Wanless, Ian M.
TI - Overlapping latin subsquares and full products
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 175
EP - 184
AB - We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac{n}{m}{n\atopwithdelims ()h}/{m\atopwithdelims ()h}$ subsquares of order $m$, where $h=\lceil (m+1)/2\rceil $. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2m}+2$ in $n$. (b) For all $n\ge 5$ there exists a loop of order $n$ in which every element can be obtained as a product of all $n$ elements in some order and with some bracketing.
LA - eng
KW - latin square; latin subsquare; overlapping latin subsquares; full product in loops; Latin square; Latin subsquare; overlapping Latin subsquare; full product in loops
UR - http://eudml.org/doc/37750
ER -