How many latin rectangles are there?

TL;DR

This paper derives a general explicit formula for counting Latin rectangles of any size, extending previous results limited to small cases, and also provides a new formula for Latin squares.

Contribution

It introduces a universal explicit formula for counting Latin rectangles of any order, generalizing prior specific cases and also deriving counts for Latin squares.

Findings

Explicit formula for Latin rectangles of any size

Unified approach encompassing previous small cases

New formula for Latin squares when m = n

Abstract

Until now the problem counting Latin rectangles m x n has been solved with an explicit formula for m = 2, 3 and 4 only. In the present paper an explicit formula is provided for the calculation of the number of Latin rectangles for any order m. The results attained up to now become particular cases of this new formula. Furthermore, putting m = n, the number of Latin squares of order n can also be obtained in an explicit form.