Minimal Center Covering Stars with Respect to LCM in Pascal’s Pyramid and its Generalizations

Abstract

Take an entry X inside m-dimensional Pascal’s pyramid consisting of m-nomial coefficients. We call a set of successive r entries starting A on the half line XA a ray of length r with center X, where A is one of the m(m + 1) entries surrounding X, and the union of nonempty set of rays with center X a star with center X. When, in the following, we assume that a star S is translatable in parallel with its center in Pascal’s pyramid, we sometimes use the same word “star” instead of “star configuration” for brevity. For a configuration of two sets of entries U and V in the pyramid, we say that U covers V w.r.t. LCM if the equality LCM U ⋃ V = LCM U always holds independent of the location of U and V as long as they are contained in the pyramid and their relative location is unchanged. We say that a star S is a center covering star w.r.t. LCM if it covers its center in the above sense, and it is minimal if it does not contain any such center covering star with the same center which is a proper subset of S.