Matches in SemOpenAlex for { <https://semopenalex.org/work/W1531263524> ?p ?o ?g. }
- W1531263524 abstract "A $k times n$ Latin rectangle $L$ is a $k times n$ array, with symbols from a set of cardinality $n$, such that each row and each column contains only distinct symbols. If $k=n$ then $L$ is a Latin square. Let $L_{k,n}$ be the number of $k times n$ Latin rectangles. We survey (a) the many combinatorial objects equivalent to Latin squares, (b) the known bounds on $L_{k,n}$ and approximations for $L_n$, (c) congruences satisfied by $L_{k,n}$ and (d) the many published formulae for $L_{k,n}$ and related numbers. We also describe in detail 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. Doyle's formula for $L_{k,n}$ is given in a closed form and is used to compute previously unpublished values of $L_{4,n}$, $L_{5,n}$ and $L_{6,n}$. We reproduce the three formulae for $L_{k,n}$ by Fu that were published in Chinese. We give a formula for $L_{k,n}$ that contains, as special cases, formulae of (a) Fu, (b) Shao and Wei and (c) McKay and Wanless. We also introduce a new equation for $L_{k,n}$ whose complexity lies in computing subgraphs of the rook's graph." @default.
- W1531263524 created "2016-06-24" @default.
- W1531263524 creator A5075468124 @default.
- W1531263524 date "2010-06-14" @default.
- W1531263524 modified "2023-10-14" @default.
- W1531263524 title "The Many Formulae for the Number of Latin Rectangles" @default.
- W1531263524 cites W139848366 @default.
- W1531263524 cites W1482309519 @default.
- W1531263524 cites W1488960947 @default.
- W1531263524 cites W1489988510 @default.
- W1531263524 cites W153541534 @default.
- W1531263524 cites W157385787 @default.
- W1531263524 cites W1576330500 @default.
- W1531263524 cites W1587704401 @default.
- W1531263524 cites W1594291410 @default.
- W1531263524 cites W1599335667 @default.
- W1531263524 cites W1602266419 @default.
- W1531263524 cites W1605824289 @default.
- W1531263524 cites W1657226914 @default.
- W1531263524 cites W1678766715 @default.
- W1531263524 cites W1695847896 @default.
- W1531263524 cites W1769440548 @default.
- W1531263524 cites W1778846403 @default.
- W1531263524 cites W1954214817 @default.
- W1531263524 cites W1965142549 @default.
- W1531263524 cites W1965237675 @default.
- W1531263524 cites W1974270301 @default.
- W1531263524 cites W1974487335 @default.
- W1531263524 cites W1974662261 @default.
- W1531263524 cites W1975997420 @default.
- W1531263524 cites W1976714507 @default.
- W1531263524 cites W1980498468 @default.
- W1531263524 cites W1981469958 @default.
- W1531263524 cites W1984221426 @default.
- W1531263524 cites W1985301313 @default.
- W1531263524 cites W1986001904 @default.
- W1531263524 cites W1988753838 @default.
- W1531263524 cites W1988904034 @default.
- W1531263524 cites W1988931620 @default.
- W1531263524 cites W1993549462 @default.
- W1531263524 cites W1997318331 @default.
- W1531263524 cites W1997967288 @default.
- W1531263524 cites W2006129849 @default.
- W1531263524 cites W2008046623 @default.
- W1531263524 cites W2008689074 @default.
- W1531263524 cites W2011164354 @default.
- W1531263524 cites W2014038201 @default.
- W1531263524 cites W2019367635 @default.
- W1531263524 cites W2020002518 @default.
- W1531263524 cites W2027584441 @default.
- W1531263524 cites W2035658154 @default.
- W1531263524 cites W2037111685 @default.
- W1531263524 cites W2038537591 @default.
- W1531263524 cites W2039481874 @default.
- W1531263524 cites W2039611200 @default.
- W1531263524 cites W2040320705 @default.
- W1531263524 cites W2040670712 @default.
- W1531263524 cites W2045276514 @default.
- W1531263524 cites W2048062291 @default.
- W1531263524 cites W2050075403 @default.
- W1531263524 cites W2050520962 @default.
- W1531263524 cites W2051010670 @default.
- W1531263524 cites W2051224597 @default.
- W1531263524 cites W2052889187 @default.
- W1531263524 cites W2058373352 @default.
- W1531263524 cites W2063117789 @default.
- W1531263524 cites W2069392999 @default.
- W1531263524 cites W2069909488 @default.
- W1531263524 cites W2073936290 @default.
- W1531263524 cites W2074039403 @default.
- W1531263524 cites W2074085132 @default.
- W1531263524 cites W2075723156 @default.
- W1531263524 cites W2079184072 @default.
- W1531263524 cites W2080249901 @default.
- W1531263524 cites W2081836846 @default.
- W1531263524 cites W2082237513 @default.
- W1531263524 cites W2082560178 @default.
- W1531263524 cites W2083140445 @default.
- W1531263524 cites W2085199361 @default.
- W1531263524 cites W2086876420 @default.
- W1531263524 cites W2088012620 @default.
- W1531263524 cites W2092653160 @default.
- W1531263524 cites W2094555811 @default.
- W1531263524 cites W2094960002 @default.
- W1531263524 cites W2105475350 @default.
- W1531263524 cites W2116353246 @default.
- W1531263524 cites W2126209209 @default.
- W1531263524 cites W2133132906 @default.
- W1531263524 cites W2142229742 @default.
- W1531263524 cites W2154750111 @default.
- W1531263524 cites W2161669520 @default.
- W1531263524 cites W2167711748 @default.
- W1531263524 cites W2168726553 @default.
- W1531263524 cites W2189119475 @default.
- W1531263524 cites W2315579714 @default.
- W1531263524 cites W2319455133 @default.
- W1531263524 cites W2319675223 @default.
- W1531263524 cites W2324748902 @default.
- W1531263524 cites W2325454248 @default.
- W1531263524 cites W2325863644 @default.