TY - JOUR AU - FU, CHIN-MEI PY - 1993/06/01 Y2 - 2024/03/28 TI - A NOTE ON THE CONSTRUCTION OF LARGE SET OF LATIN SQUARES WITH ONE ENTRY IN COMMON JF - Tamkang Journal of Mathematics JA - Tamkang J. Math. VL - 24 IS - 2 SE - Papers DO - 10.5556/j.tkjm.24.1993.4492 UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4492 SP - 215-220 AB - <div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">A latin square of order $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">is an $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n \times</span> <span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">array such that each of the integers $1, 2, 3, \cdots, </span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">occurs exactly once in each row and each column. A large set of latin squares of order $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">having only one entry in common is a maximum set of latin squares of order $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">such that each pair of them contains exactly one fixed entry in common. In this paper, we prove that a large set of latin squares of order $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">having only one entry in common has $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n - </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">1$ latin squares for each positive integer $</span><span style="font-size: 11.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$, $n </span><span style="font-size: 12.000000pt; font-family: 'TimesNewRomanPSMT';">\ge </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">4$. </span></p></div></div></div> ER -