The Erdös–Szekeres tableau of a permutation is the sequence of points where (respectively ) is the length of the longest increasing (respectively decreasing) subsequence ending at . Since different permutations can have the same Erdös–Szekeres tableau (EST) (e.g. and both have the same "N-shaped" EST), there is an equivalence relation on permutations . The poset is defined by taking the intersection over all orderings induced by elements of . Informally, the poset records those relations that can be recovered from the EST. The lattice is defined on , where is in the covering relation if and differ by an adjacent transposition (which can be viewed as an edge label) and precedes lexicographically.
This Demonstration illustrates concepts developed in . The green curve in the first pane visualizes a permuatation by plotting the points for .
 S. V. Ault and B. Shemmer, "Erdös–Szekeres Tableaux," Order, 13, 2013. doi: 10.1007/s11083-013-9308-2.