Divisors of the number of Latin rectangles

A kxn Latin rectangle on the symbols {1,2,...,n} is called reduced if the first row is (1,2,...,n) and the first column is (1,2,...,k)^T. Let R"k","n be the number of reduced kxn Latin rectangles and [email protected]?n/[email protected]?. We prove several results giving divisors of R"k","n. For example, (k-1)! divides R"k","n when k=

[1]  R. A. Bailey.  Latin squares with highly transitive automorphism groups , 1982 .

[2]  Michael J. Mossinghoff,et al.  Combinatorics and graph theory , 2000 .

[3]  Gary L. Mullen,et al.  How Many i-j Reduced Latin Rectangles are There? , 1980 .

[4]  N. Yu. Kuznetsov,et al.  Estimating the number of latin rectangles by the fast simulation method , 2009 .

[5]  Arthur A. Drisko.  Proof of the Alon-Tarsi Conjecture for n=2rp , 1998, Electron. J. Comb..

[6]  I. Gessel.  Counting three-line Latin rectangles , 1986 .

[7]  Navin M. Singhi,et al.  On the Number of Latin Rectangles and Chromatic Polynomial of L(Kr, s) , 1980, Eur. J. Comb..

[8]  S. M. Jacob.  The Enumeration of the Latin Rectangle of Depth Three by Means of a Formula of Reduction, with other Theorems Relating to Non‐Clashing Substitutions and Latin Squares , 2022 .

[9]  Ian M. Wanless,et al.  On the Number of Latin Squares , 2005, 0909.2101.

[10]  Gary L. Mullen,et al.  How Many i-j Reduced Latin Rectangles are There? , 1980 .

[11]  Gary L. Mullen,et al.  Enumeration formulas for latin and frequency squares , 1993, Discret. Math..

[12]  John Riordan.  A Recurrence Relation for Three-Line Latin Rectangles , 1952 .

[13]  Timothy A. Green.  Asymptotic enumeration of generalized latin rectangles , 1989, J. Comb. Theory, Ser. A.

[14]  J. Q. Longyear,et al.  Counting $3$ by $n$ Latin rectangles , 1976 .

[15]  Brendan D. McKay,et al.  Latin Squares of Order 10 , 1995, Electron. J. Comb..

[16]  J. Riordan.  Three-Line Latin Rectangles—II , 1944 .

[17]  Jia-Yu Shao,et al.  A formula for the number of Latin squares , 1992, Discret. Math..

[18]  Koichiro Yamamoto,et al.  An Asymptotic Series for the Number of Three-Line Latin Rectangles , 1950 .

[19]  Koichi Yamamoto.  Symbolic methods in the problem of three-line Latin rectangles , 1953 .

[20]  Arthur A. Drisko.  On the Number of Even and Odd Latin Squares of Orderp+1 , 1997 .

[21]  D. C. van Leijenhorst.  Van Der Corput's "scriptum 3"" , 2000 .

[22]  Paul Erdös,et al.  The Asymptotic Number of Latin Rectangles , 1946 .

[23]  J. Shaw.  Combinatory Analysis , 1917, Nature.

[24]  A.-A. A. Jucys,et al.  The Number of Distinct Latin Squares as a Group-Theoretical Constant , 1976, J. Comb. Theory, Ser. A.

[25]  C. R. Pranesachar,et al.  Enumeration of Latin rectangles via SDR’s , 1981 .

[26]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[27]  L. Carlitz.  Congruences connected with three-line latin rectangles , 1953 .

[28]  Gary L. Mullen.  How many i-j reduced Latin squares are there? Amer , 1978 .

[29]  Ronald Alter.  How Many Latin Squares are There , 1975 .

[30]  I. Gessel.  Counting Latin rectangles , 1987 .

[31]  B. McKay,et al.  Small latin squares, quasigroups, and loops , 2007 .

[32]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[33]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[34]  Brendan D. McKay,et al.  Asymptotic enumeration of Latin rectangles , 1984 .

[35]  P. Doyle.  The number of Latin rectangles , 2007, math/0703896.