Scholarlink

<p>The FKG inequality is a powerful tool for proving inequalities in distributive lattices. We show how a special case, which we call the Order Ideal Lemma, can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving lattice paths (Catalan, Motzkin and large Schröder numbers), intervals in Young’s lattice, order polynomials, specializations of Schur and Schur <i>Q</i>-functions, Lucas sequences, descent and peak polynomials of permutations, pattern avoidance, set partitions, and noncrossing partitions. We end with a section with conjectures and outlining future directions.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Log-Concavity and Log-Convexity via Distributive Lattices

  • 期刊论文

  • 发表日期：

    2026年5月2日

查看全文
  • Jinting Liang,
  • Bruce E. Sagan

摘要

The FKG inequality is a powerful tool for proving inequalities in distributive lattices. We show how a special case, which we call the Order Ideal Lemma, can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving lattice paths (Catalan, Motzkin and large Schröder numbers), intervals in Young’s lattice, order polynomials, specializations of Schur and Schur Q-functions, Lucas sequences, descent and peak polynomials of permutations, pattern avoidance, set partitions, and noncrossing partitions. We end with a section with conjectures and outlining future directions.