The aim of this talk is to show how techniques from the formal logic world can sometimes be applied directly to specific mathematical problems studied completely independently in the world of combinatorics.

One basic activity in combinatorics is to establish combinatorial identities by so-called `bijective proofs,' which consist of constructing explicit bijections between two types of the combinatorial objects under consideration.

Based on the author's two-directional multiset rewriting technique and reasonably `restricted' two-directional strong normalization, recently I have fully characterized all equinumerous partition ideals whose complementary order filters are generated by pairwise disjoint minimal elements, with providing, in addition, the most `natural bijections' between the partition ideals.

Along the above transparent `geometrical' lines, I will show a much stronger bijection form of Andrews's and Subbarao's results on so-called `Euler pairs', and, moreover, I will fully characterize the class of all Euler-like partition identities, in which the number of repeated parts is variable and controlled by some function $f$.

In the second, "overlapping", part of my talk, I discuss the challenges of the `overlapping' multiset rewriting systems. I will show the `bijective proofs' (based on the two-directional multiset rewriting technique indeed) of a new series of partition identities related to Fibonacci and Lucas numbers.

In closing, I will show a much stronger form of the above `bijective proofs' - that the related fine partition statistics have identical distribution functions.

No acquaintance with partitions, etc. is required.

For more information about the Penn Logic and Computation Seminar, please see the seminar web page.