Comparing increasing ordered statistical data represented by sequences of positive integers

Given two types of ordered statistical discrete data in the form 1 ≤ i₁ ≤ i₂ ≤ … ≤ i_n ≤ n where i_j = 1,2,…,n, j =1,2,…,n. The first type of these data is the order statistics of a sample of size n taken from the discrete uniform distribution on S={1,2,…,n}. The other type of these sequences arise in some practical situations. We study the property that i_j = j, j = 1,2,…, n in each type and obtain the probability distribution functions as well as the asymptotic distributions of the random variables representing this property. The result can be used to distinguish between these data.