OEIS A000186: 3×N Latin Rectangles with an Ordered First Row

A deep dive into A000186, the count of 3×N Latin rectangles with the first row fixed in increasing order. We unpack what a Latin rectangle is, explain what 'ordered first row' means (reduced rectangles), and trace the history from McMahon’s framework to Riordan’s rigorous 3-row proof. We then explore Erdos and Kaplansky’s sieve-based asymptotics for growing numbers of rows, the refinements for the 3×N case (including numerical evidence by Karawala), and the major extension by Godsil and McKay that widens the range of valid k. Along the way we connect the asymptotic formula roughly f(n,k) ~ (n!)^k exp(-k(k−1)/2) to the specific 3×N case, highlighting how these ideas fit into broader combinatorics and the OEIS landscape.

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