Level Matrices

الأصل · EN

Let n>1 and k>0 be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with m rows is called reducible if we can delete j rows, 0<j<m, so that the remaining matrix is level. We ask if there is a minimum integer ℓ=ℓ(n,k) such that for all m>ℓ, any m× n level matrix with entries in {0,,k} is reducible. It is known that ℓ(2,k)=2k-1. In this paper, we establish the existence of ℓ(n,k) for n≥ 3 by giving upper and lower bounds for it. We then apply this result to bound the number of certain types of vector space multipartitions.