This question is cross-posted from math.stackexchange because it might be too technical. Let $S_n$ be the symmetric group. Recall that the number of inversions of a permutation $\ …

Let $n,k$ be positive integers such that $3n=2k$ and $N = \lfloor \alpha n\rfloor$ for some constant $0<\alpha<1$. Let $S_{3n}$ denote the permutation group of order $3n$. Co …

This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated …

Suppose $ S_{n,N} $ be the set of $n$ elements with $N$ many cycles where $N$ is proportional to $n$. $U_{n,N}$ is an element picked randomly from this. It is known that the length …

Given n numbers (each of which is a random integer, uniformly between 1~n), what is the expected number of increasing subsequences?