Erdos Szekeres Theorem talks about the presence of a monotonically increasing or decreasing subsequence in a sequence of fixed length. However, in this work we are interested in the number of such sequences given permutation length n.
In this work, we introduce a new theorem for minimum number of distinct monotonically increasing or decreasing subsequence of length n+1 in a permutaion of length k, given n and k. Additionally, we present a conjecture for upper bound on the number of distinct monotonic sequences of length n+1 in sequence of length l.