The interval structure of (0,1)-matrices

Abstract

Let A be an n × n (0, * )-matrix, so each entry is 0 or * . An A-interval matrix is a (0, 1)-matrix obtained from A by choosing some * 's so that in every interval of consecutive * 's, in a row or column of A, exactly one * is chosen and replaced with a 1, and every other * is replaced with a 0. We consider the existence questions for A-interval matrices, both in general, and for specific classes of such A defined by permutation matrices. Moreover, we discuss uniqueness and the number of A-permutation matrices, as well as properties of an associated graph.

Multi-alternating sign matrices

“…Here, k * = k * (10) = 10/3 = 4, so the matrix A (4,10) above has the maximum number of nonzeros among 2-regular ASMs of order 10. Additional information concerning dense ASMs is in [11].…”

“…Not every 2-regular ASM results in this way from an ASM. The example of a 2-regular ASM with the maximum number of nonzeros when n = 10, the matrix A (4,10) given in Example 3.5, has 68 nonzeros while the maximum number of nonzeros of a ASM with n = 10 is 50 (the diamond ASM); but 68 − 50 = 18 = 10. For example,…”