Erdos-Szekeres tableaux

TL;DR

This paper introduces the Erdős-Szekeres tableau (EST), a geometric representation of sequences that captures the lengths of increasing and decreasing subsequences, and investigates what sequence information can be recovered from it.

Contribution

The paper develops the concept of EST and the Order Poset to analyze sequence properties and recoverability, offering a new geometric approach to the Erdős-Szekeres problem.

Findings

EST encodes key subsequence length information

Order Poset helps determine sequence recoverability

Geometric approach offers new insights into classical combinatorial problems

Abstract

We explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometric approach to answer it. Our main object of study is the Erd\H{o}s-Szekeres tableau, or EST, of a number sequence. An EST is the sequence of integral points whose coordinates record the length of the longest increasing and longest decreasing subsequence ending at each element of the sequence. We define the Order Poset of an EST in order to answer the question: What information about the sequence can be recovered by its EST?