Erdős–Szekeres intersection property visualization

If the product of the lengths of the longest increasing subsequence (LIS) and longest decreasing subsequence (LDS) of a permutation equals the length of the permutation (a·b = k), then one can choose an LIS of length a and an LDS of length b that share a common element. When a·b = k, every pair of labels (a_i,b_i) appears exactly once. A suitable LIS is obtained by taking elements with b_i=1 in order of increasing a_i; a suitable LDS is obtained by taking elements with a_i=1 in order of increasing b_i. These two subsequences intersect at the element labelled (1,1).

Choose k (sequence length): 6

Intersection property for LIS

Intersection property for LDS

Intersection

Other

Index	Value	LIS length (a_i)	LDS length (b_i)