# Erdös-Szekeres Tableaux

Erdös-Szekeres Tableaux

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.

(σ)

σ=(,…,)

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1

a

n

n

{(,)}

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y

i

i=1

x

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y

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a

i

(2,1,4,3)

(3,1,4,2)

[σ]={σ~τ|(σ)=(τ)}

[σ]

[σ]

(τ,τ')

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τ'

τ

τ'