Matrices uniquely determined by their lonesums

A matrix is lonesum if it can be uniquely reconstructed from its row and column sums. Brewbaker computed the number of m×n binary lonesum matrices. Kaneko defined the poly-Bernoulli numbers of an integer index, and showed that the number of m×n binary lonesum matrices is equal to the mth poly-Bernoulli number of index -n. In this paper, we are interested in q-ary lonesum matrices. There are two types of lonesumness for q-ary matrices, namely strongly and weakly lonesum. We first study strongly lonesum matrices: We compute the number of m×n q-ary strongly lonesum matrices, and provide a generalization of Kaneko’s formulas by deriving the generating function for the number of m×n q-ary strongly lonesum matrices. Next, we study weakly lonesum matrices: We show that the number of forbidden patterns for q-ary weakly lonesum matrices is infinite if q⩾5, and construct some forbidden patterns for q=3,4. We also suggest an open problem related to ternary and quaternary weakly lonesum matrices.