Advanced determinant calculus.
Let be a non-empty subset of positive integers. A partition of a positive integer into is a finite nondecreasing sequence of positive integers in with repetitions allowed such that . Here we apply Pólya’s enumeration theorem to find the number of partitions of into , and the number of distinct partitions of into . We also present recursive formulas for computing and .
We present a new proof of Whitney's broken circuit theorem based on induction on the number of edges and the deletion-contraction formula.