Counting Three-Line Latin Rectangles

A k × n Latin rectangle is a k × n array of numbers such that (i) each row is a permutation of [n] = {1, 2, . . . , n} and (ii) each column contains distinct entries. If the first row is 12 · · ·n, the Latin rectangle is said to be reduced. Since the number k × n Latin rectangles is clearly n! times the number of reduced k× n Latin rectangles, we shall henceforth consider only reduced Latin rectangles. It is known [7, exercise 4.5.10, p. 288; solution, p. 507] that the number of (reduced) 3 × n Latin rectangles is the coefficient of x/n! in e ∞ ∑