Weighted K-k-Schur functions and their application to the K-k-Schur alternating conjecture

We introduce the new concept of weighted K-k-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both K-k-Schur functions and closed k-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed k-bounded partitions. As a central application, we resolve the K-k-Schur alternating conjecture -- posed by Blasiak, Morse, and Seelinger in 2022 -- for a wide class of k-bounded partitions, including all strictly decreasing k-bounded partitions. Our results shed new light on the combinatorial structure of K-theoretic symmetric functions.