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Let a, b, c, d, e be positive integers such that abcde. 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.


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