# Overlapping latin subsquares and full products

• Volume: 51, Issue: 2, page 175-184
• ISSN: 0010-2628

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## Abstract

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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 $\frac{n}{m}\left(\genfrac{}{}{0pt}{}{n}{h}\right)/\left(\genfrac{}{}{0pt}{}{m}{h}\right)$ subsquares of order $m$, where $h=⌈\left(m+1\right)/2⌉$. 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.

## How to cite

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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 -

## References

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1. Dénes J., Hermann P., On the product of all elements in a finite group, Ann. Discrete Math. 15 (1982), 105–109. MR0772587
2. Heinrich K., Wallis W.D., 10.1007/BFb0091822, Lecture Notes in Math. 884 (1981), 221–233. Zbl0475.05014MR0641250DOI10.1007/BFb0091822
3. McKay B.D., Wanless I.M., 10.1006/jcta.1998.2947, J. Combin. Theory Ser. A 86 (1999), 323–347. Zbl0948.05014MR1685535DOI10.1006/jcta.1998.2947
4. Pula K., Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture, Electron. J. Combin. 16 (2009), R57. MR2505099
5. Ryser H.J., 10.1090/S0002-9939-1951-0042361-0, Proc. Amer. Math. Soc. 2 (1951), 550–552. Zbl0043.01202MR0042361DOI10.1090/S0002-9939-1951-0042361-0
6. van Rees G.H.J., Subsquares and transversals in latin squares, Ars Combin. 29B (1990), 193–204. Zbl0718.05014MR1412875

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