Enumeration and classification of self-orthogonal partial Latin rectangles by using the polynomial method
Raúl Manuel Falcón Ganfornina · Scuola Normale Superiore eBooks · 2013
The current paper deals with the enumeration and classification of the set SOPLR r,n of self-orthogonal r × r partial Latin rectangles based on n symbols. It can be identified with the set of zeros of the zero-dimensional ideal I r,n = (x ijk · (1 − x ijk ), x ijk · x i′jk , x ijk · x ij′k , x ijk · xijk′: i ϵ [r], j ϵ [s], k ϵ [n], i′ ϵ [r] - [i], j′ ϵ [s] - [j], k′ ϵ [n] - [k] 〉 ∪ 〈 x ijp · x klp · x jiq · x lkq:i, j, k,l ϵ [r], p,q ϵ [n], (i, j) ≠ (k, l) 〉 ⊆ ℚ[x 111, … x rrn ]. Moreover, |SOPLRx r,n | = dimℚ(ℚ[x 111, …, x rrn ]/I –r,n).