Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method

The current paper deals with the enumeration and classification of the set SORr,nSORr,n of self-orthogonal r×rr×r partial Latin rectangles based on nn symbols. These combinatorial objects are identified with the independent sets of a Hamming graph and with the zeros of a radical zero-dimensional ideal of polynomials, whose reduced Gröbner basis and Hilbert series can be computed to determine explicitly the set SORr,nSORr,n. In particular, the cardinality of this set is shown for r≤4r≤4 and n≤9n≤9 and several formulas on the cardinality of SORr,nSORr,n are exposed, for r≤3r≤3. The distribution of r×sr×s partial Latin rectangles based on nn symbols according to their size is also obtained, for all r,s,n≤4r,s,n≤4.