Let a, b, c, d, e be positive integers such that a ≤ b ≤ c ≤ d ≤ e. Heinrich showed the existence of a partitioned incomplete Latin square (PILS) of type (a, b, c) and (a, b, c, d) if and only if a = b = c and 2a ≥ d. For PILS of type (a,b,c,d,e), it is necessary that a + b + c ≥ e, but not suﬃcient, and no characterization is currently known. In this talk we provide an additional necessary condition, classify the existence of PILS of type (a, b, c, d, a + b + c) and PILS with three equal parts, and show the existence of a family of PILS in which the parts are nearly the same size.
J. Kuhl, D. G. McGinn and M.W. Schroeder, On the existence of partitioned incomplete Latin squares with five parts. Australas. J. Combin. 74 (2019), 46–60.