Further results on large sets of partitioned incomplete Latin squares

In this article, we continue to study the existence of large sets of partitioned incomplete Latin squares (LSPILS). We complete the determination of the spectrum of an LSPILS ( g n ) and prove that there exists an LSPILS ( g n ) if and only if g ≥ 1 , n ≥ 3 , and ( g , n ) ≠ ( 1 , 6 ) . We also start the investigation of LSPILS with two group sizes and prove that there exists an LSPILS + ( g n( 2 g ) 1 ) for all g ≥ 1 and n ≥ 3 except possibly for n ≡ 2 , 10( mod 12 ) and n ≥ 14 . Furthermore, we obtain a pair of orthogonal LSPILS + ( 1 p 2 1 ) s, where p is a prime and p ≡ 1( mod 6 ) .