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Brun's inequality for a geometric lattice

M. Ram Murty, Sunil Naik

Abstract

In a seminal paper of 1915, V. Brun introduced Brun's sieve, which is based on Brun's inequality for the Möbius function and is a very powerful tool in modern number theory. The importance of the Möbius function in enumeration problems led G.-C. Rota to introduce the concept of the Möbius function to partially ordered sets. In this article, we prove Brun's inequality for geometric lattices and develop a sieve in this context. One of the main ingredients is a recent work of K. Adiprasito, J. Huh, and E. Katz on the log-concavity of absolute values of the Whitney numbers associated with matroids. We also study shifted convolutions of the Whitney numbers associated with Dowling lattices. Further, we derive an asymptotic formula for generalized Dowling numbers.

Brun's inequality for a geometric lattice

Abstract

In a seminal paper of 1915, V. Brun introduced Brun's sieve, which is based on Brun's inequality for the Möbius function and is a very powerful tool in modern number theory. The importance of the Möbius function in enumeration problems led G.-C. Rota to introduce the concept of the Möbius function to partially ordered sets. In this article, we prove Brun's inequality for geometric lattices and develop a sieve in this context. One of the main ingredients is a recent work of K. Adiprasito, J. Huh, and E. Katz on the log-concavity of absolute values of the Whitney numbers associated with matroids. We also study shifted convolutions of the Whitney numbers associated with Dowling lattices. Further, we derive an asymptotic formula for generalized Dowling numbers.

Paper Structure

This paper contains 17 sections, 7 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Statements of Results
  3. Preliminaries
  4. Unimodality and log-concavity
  5. Prerequisites from theory of partially ordered sets (posets)
  6. Prerequisites from theory of matroids
  7. Prerequisites from Dowling lattices
  8. Proofs of main theorems
  9. Proof of Theorem \ref{['ThmU']}
  10. Proof of Theorem \ref{['ThmB']}
  11. A sieve on a geometric lattice
  12. Proof of Theorem \ref{['thmSift']}
  13. Sifted sets and Shifted convolution of Whitney numbers
  14. Proof of Proposition \ref{['propshiftW']}
  15. Sifted sets and Dowling numbers
  16. ...and 2 more sections

Key Result

Theorem 1

Let $L$ be a geometric lattice. Then for any positive integer $k$, we have Here $\mu$ denotes the Möbius function on $L$ and $r$ denotes a rank function on $L$.

Theorems & Definitions (9)

  • Theorem 1: Brun's inequality for a geometric lattice
  • Theorem 2
  • Theorem 3
  • Remark 2.1
  • Proposition 4
  • Theorem 5
  • Conjecture : Heron-Rota-Welsh
  • Theorem 6
  • Theorem 7