The number of Latin rectangles

TL;DR

This paper derives a general expression for counting k-line Latin rectangles, revealing its computational complexity and extending Ryser's formula for derangements to a broader combinatorial context.

Contribution

It introduces a new formula for the number of k-line Latin rectangles applicable for any k, expanding the mathematical tools for combinatorial enumeration.

Findings

Expression for counting k-line Latin rectangles derived

Computational complexity is on the order of n^(2^(k-1))

Generalizes Ryser's formula for derangements

Abstract

We show how to generate an expression for the number of k-line Latin rectangles for any k. The computational complexity of the resulting expression, as measured by the number of additions and multiplications required to evaluate it, is on the order of n^(2^(k-1)). These expressions generalize Ryser's formula for derangements.