## Clustered k-Subsets

July 11, 2010 at 04:39 (algorithm, java, math.CO)

This post is mostly copied and pasted from my posts in this thread on MHF.

Problem statement: Let $\displaystyle S_n = \{1,\ \dots\ ,n\}$ and let $\displaystyle f(n,k,x)$ be the number of k-subsets of $\displaystyle S_n$ such that when the elements are ordered from least to greatest, the absolute difference of any two adjacent elements is less than $\displaystyle x$. Find a formula or algorithm to determine $\displaystyle f(n,k,x)$ for arbitrary $\displaystyle n,k,x \in \mathbb{Z}, n\ge 0$.