Enumeration of Truncated Latin Rectangles
F. W. Light · The Fibonacci Quarterly · 1979
An r xk rectangle is a rectangular array of elements (natural numbers) with r rows and k columns. A row with no repeated element is an R-row. A column with no repeated element is a (7-column; otherwise, it is a C-column. If all rows of a rectangle are i?-rows, it is an R-rectangle. An i?-rectangle subject to no further restrictions will be called, for emphasis, free. One whose first row is prescribed (elements arranged in increasing numerical order) is a normalized i?-rectangle. An i?-rectangle all of whose columns are C-columns is an R-C-rectangle; one whose columns are all C-columns is an R-C-rectangle. An r xn R-C-rectangle each of whose rows consists of the same n elements is a Latin rectangle (L-rectangle). (i?-C-rectangles whose, rows do not all consist of the same elements are the "truncated " L-rectangles of the title.) ENUMERATION OF CERTAIN i?-RECTANGLES The most obvious enumerational question about L-rectangles is, probably: How many distinct normalized r xn L-rectangles are there? Denoting this number as M*9 we have, as in [1], (1) M: = ±(-l)^[(n-k)lf-\\ k = 0 where & r>n is the number of free r xk R-C-vec tangles that can be built up with (7-columns constructed from elements selected from r rows each of which consists of the elements 1, 2,..., n. The number of free r xn L-rectangles is (2) ^ = £(-l) S (;!)[(n- 0 and all n. k