Hall's theorem and extending partial latinized rectangles

We generalize a theorem of M. Hall Jr., that an r×nr×n Latin rectangle on n   symbols can be extended to an n×nn×n Latin square on the same n symbols. Let p, n  , ν1,ν2,…,νnν1,ν2,…,νn be positive integers such that 1≤νi≤p1≤νi≤p(1≤i≤n)(1≤i≤n) and ∑i=1nνi=p2. Call an r×pr×p matrix on n   symbols σ1,σ2,…,σnσ1,σ2,…,σn an r×pr×p(ν1,ν2,…,νn)(ν1,ν2,…,νn)-latinized rectangle   if no symbol occurs more than once in any row or column, and if the symbol σiσi occurs at most νiνi times altogether (1≤i≤n)(1≤i≤n). We give a necessary and sufficient condition for an r×pr×p(ν1,ν2,…,νn)(ν1,ν2,…,νn)-latinized rectangle to be extendible to a p×pp×p(ν1,ν2,…,νn)(ν1,ν2,…,νn)-latinized square. The condition is a generalization of P. Hall's condition for the existence of a system of distinct representatives, and will be called Hall's (ν1,ν2,…,νn)(ν1,ν2,…,νn)-Constrained Condition. We then use our main result to give two further sets of necessary and sufficient conditions. Finally we use our results to show that, given p, n  , ν1,ν2,…,νnν1,ν2,…,νn such that 1≤νi≤p1≤νi≤p, ∑i=1nνi=p2, then a p×pp×p(ν1,ν2,…,νn)(ν1,ν2,…,νn)-latinized square exists.