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

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

  • 期刊论文

  • 发表日期：

    2026年5月9日

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  • Yaozhou Fang,
  • Xing Gao

摘要

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.