Friendly paths for finite subsets of plane integer lattice. I

TL;DR

This paper characterizes all finite point configurations in the 2D integer lattice that cannot be separated by a friendly path, providing new sequences and solving a longstanding problem in lattice point separation.

Contribution

It fully describes inseparable point sets in Z^2 and introduces two new sequences to OEIS, solving a problem from the American Mathematical Monthly.

Findings

Exactly c(n) inseparable sets up to lattice symmetry.

Sequences c(n) and (n) are new entries in OEIS.

Inseparable sets exist for all even n2 and almost all odd n.

Abstract

For a given finite subset P of points of the lattice Z^2, a friendly path is a monotone (uphill or downhill) lattice path which splits points in half; points lying on the path itself are discarded. The purpose of this paper (and its sequel) is to fully describe all configurations of n points in Z^2 which do not admit a friendly path. We say that such an n-set is inseparable. There are, up to the lattice symmetry, exactly c(n) such sets. If only lattice shifts are counted, there are \^c(n) of them. Both sequences are new entries into OEIS (A369382 and, respectively, A367783). In particular, n=27 is the first odd numbers with c(n)=1. No example was known so far. This solves problem 11484(b)* posed in American Mathematical Monthly (February 2010). In this paper we also show that inseparable n-set exist for all even numbers n>=12 and almost all odd numbers.