A classification of nonzero skew immaculate functions

This article presents conditions under which the skewed version of immaculate noncommutative symmetric functions are nonzero. The work is motivated by the quest to determine when the matrix definition of a skew immaculate function aligns with the Hopf algberaic definition. We describe a necessary condition for a skew immaculate function to include a non-zero term, as well as a sufficient condition for there to be at least one non-zero term that survives any cancellation. We bring in several classical theorems such as the Pigeonhole Principle from combinatorics and Hall's Matching Theorem from graph theory to prove our theorems.