PIE-sums: A combinatorial tool for partition theory

Magnetic Resonance Imaging: A Tool for Pork Pie Development

The traditional British pork pie consists of roughly chopped pork cooked in a hot water pastry crust. Due to shrinkage of the meat during cooking, the gap formed around the meat is usually sealed using a gelatin based jelly to exclude air and thus help to preserve the pie. The properties of the jelly are such that it will ingress into the pastry crust causing undesirable softening. The jelly is...

Dedekind sums: a combinatorial-geometric viewpoint

A Combinatorial Proof on Partition Function Parity

One of the most basic results on the number-theoretic properties of the partition function p(n) is that p(n) takes each value of parity infinitely often. First proved by Kolberg in 1959, this statement was strengthened by Kolberg and Subbarao in 1966 to say that both p(2n) and p(2n + 1) take each value of parity infinitely often. These results have received several other proofs, each relying to...

Left-inversion of combinatorial sums

The inversion of combinatorial sums is a fundamental problem in algebraic combinatorics. Some combinatorial sums, such as a n = P k d n;k b k , can not be inverted in terms of the orthogonality relation because the innnite, lower triangular array P = fd n;k g's diagonal elements are equal to zero (except d 0;0). Despite this, we can nd a left-inverse P such that P P = I and therefore are able t...

Combinatorial sums and finite differences

We present a new approach to evaluating combinatorial sums by using finite differences. Let {ak}k=0 and {bk}k=0 be sequences with the property that ∆bk = ak for k ≥ 0. Let gn = ∑n k=0 ( n k ) ak, and let hn = ∑n k=0 ( n k ) bk. We derive expressions for gn in terms of hn and for hn in terms of gn. We then extend our approach to handle binomial sums of the form ∑n k=0 ( n k ) (−1)ak, ∑ k ( n 2k ...