Let $X=(X_1,\ldots,X_n)$ be a vector of i.i.d. random variables where $X_i$'s take their values over $\mathbb{N}$. The purpose of this paper is to study the number of weakly increasing subsequences of $X$ of a given length $k$, and the number of all weakly increasing subsequences of $X$. For the former of these two, it is shown that a central limit theorem holds. Also, the first two moments of each of these two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a similarity measure of Steele, and on increasing subsequences of riffle shuffles. 查看全文>>