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Several New Generalizations of LYM Inequality

Zihao Huang, Weikang Liang, Yujiao Ma, Suijie Wang

Abstract

The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of $r$-decompositions, where all components are required to avoid chains of the same length. In this paper, we relax this constraint by allowing components of a family of $r$-decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, $q$-analog, continuous analog, and arithmetic analog frameworks. Notably, the bound in our LYM inequalities does not depend on the maximal length of all forbidden chains. Moreover, we extend our approach beyond $r$-decompositions to $r$-multichains, and establish analogous LYM inequalities.

Several New Generalizations of LYM Inequality

Abstract

The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of -decompositions, where all components are required to avoid chains of the same length. In this paper, we relax this constraint by allowing components of a family of -decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, -analog, continuous analog, and arithmetic analog frameworks. Notably, the bound in our LYM inequalities does not depend on the maximal length of all forbidden chains. Moreover, we extend our approach beyond -decompositions to -multichains, and establish analogous LYM inequalities.

Paper Structure

This paper contains 6 sections, 10 theorems, 102 equations, 2 tables.

Table of Contents

  1. Background
  2. Main results
  3. Generalizations of the set-theoretic LYM inequality
  4. Generalizations of the $q$-analog of the LYM inequality
  5. Generalizations of the continuous analog of the LYM inequality
  6. Generalizations of the arithmetic analog of the LYM inequality

Key Result

Theorem 3.1

Let $t_1, \ldots, t_r$ be positive integers and $\sigma = \frac{t_1t_2\cdots t_r}{\mathrm{max}\{t_1, \ldots, t_r\}}$. Suppose $\mathcal{D}$ is a family of $r$-decompositions of $[n]$ such that $\mathcal{D}_k$ is $t_k$-chain free for each $k \in \{1, \ldots, r\}$. Then Consequently, where $m_1,\ldots,m_\sigma$ are the $\sigma$ largest $\binom{n}{{a_1, \ldots, a_r}}$ for non-negative integers $a_1

Theorems & Definitions (23)

  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['Dec']}
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • proof : Proof of theorem \ref{['Ch']}
  • Theorem 4.1
  • ...and 13 more