Lattice points and simultaneous core partitions

TL;DR

This paper links simultaneous (a,b)-core partitions to lattice points in rational simplices, applying Ehrhart and Euler-Maclaurin theories to prove key combinatorial formulas and conjectures, and exploring connections to rational Catalan combinatorics.

Contribution

It introduces a geometric framework for (a,b)-core partitions, providing new proofs of known formulas and conjectures, and extends the analysis to rational Catalan structures and related statistics.

Findings

Reproved Anderson's theorem on the count of (a,b)-cores.

Proved Armstrong's conjecture on average size of (a,b)-cores.

Connected conjectures to Catalan hyperplane arrangements and permutation statistics.

Abstract

We observe that for a and b relatively prime, the "abacus construction" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate (a,b)-cores. We conjecture a unimodality result for q rational Catalan numbers, and make preliminary investigations in applying these methods to the (q,t)-symmetry and specialization conjectures. We prove these conjectures for low degree terms and when a=3, connecting them to the Catalan hyperplane arrangement and quadratic permutation statistics.