Problem Statement
Give an asymptotic formula for the number of $k\times n$ Latin rectangles.
Categories:
Combinatorics
Progress
Erdős and Kaplansky [ErKa46] proved the count is\[\sim e^{-\binom{k}{2}}(n!)^k\]when $k=o((\log n)^{3/2-\epsilon})$. Yamamoto [Ya51] extended this to $k\leq n^{1/3-o(1)}$.The count of such Latin rectangles is A001009 in the OEIS.
Source: erdosproblems.com/725 | Last verified: January 16, 2026