Erdős-Szekeres Tableaux

We explore a question related to the celebrated Erdős-Szekeres Theorem and develop a geometric approach to answer it. Our main object of study is the Erdős-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?