Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Foata, Hikita, and the Bulldozer Problem

Timothy Y. Chow

Abstract

In a remarkable paper, Tatsuyuki Hikita settled a longstanding e-positivity conjecture of Stanley and Stembridge. Among many other things, he wrote down a certain formula ${\varphi}_k$, and proved that the ${\varphi}_k$ sum to one, thereby defining a probability distribution. Though Hikita's proof was simple, it remains surprising that the ${\varphi}_k$ sum to one. In this note, we give a combinatorial interpretation of Hikita's probability distribution. The main tool is a certain permutation statistic that we call the watershed. After seeing an early version of our work, Darij Grinberg noticed that the permutation statistic was implicit in a so-called "bulldozer problem" that was on the short list for the 2015 International Mathematics Olympiad. However, our description of the statistic, which makes use of the Renyi-Foata bijection, appears to be new.

Foata, Hikita, and the Bulldozer Problem

Abstract

In a remarkable paper, Tatsuyuki Hikita settled a longstanding e-positivity conjecture of Stanley and Stembridge. Among many other things, he wrote down a certain formula , and proved that the sum to one, thereby defining a probability distribution. Though Hikita's proof was simple, it remains surprising that the sum to one. In this note, we give a combinatorial interpretation of Hikita's probability distribution. The main tool is a certain permutation statistic that we call the watershed. After seeing an early version of our work, Darij Grinberg noticed that the permutation statistic was implicit in a so-called "bulldozer problem" that was on the short list for the 2015 International Mathematics Olympiad. However, our description of the statistic, which makes use of the Renyi-Foata bijection, appears to be new.
Paper Structure (5 sections, 4 theorems, 12 equations)

This paper contains 5 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. The Permutation Statistic
  3. Hikita's Transition Probabilities
  4. The Bulldozer Problem
  5. Acknowledgments

Key Result

Theorem 1

Let $\pi_1,\pi_2,\ldots, \pi_{2n}$ be a sequence of $2n$ distinct integers. Then there exists a unique $k\in\{0,1,\ldots,n\}$ such that all the cycle lengths of are even.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:watershed']}
  • Lemma 2
  • proof
  • Theorem 2
  • proof