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Friendly paths for finite subsets of plane integer lattice. I

Giedrius Alkauskas

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 ĉ(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.

Friendly paths for finite subsets of plane integer lattice. I

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 ĉ(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.
Paper Structure (16 sections, 5 theorems, 21 equations, 15 figures, 1 table)

This paper contains 16 sections, 5 theorems, 21 equations, 15 figures, 1 table.

Table of Contents

  1. Friendly paths
  2. Formulation
  3. Results
  4. Preliminary quartering
  5. Representations of inseparable sets
  6. The structure
  7. Examples
  8. Full conditions for inseparability
  9. Evenness and proper quartering
  10. Example with $n=108$
  11. General check
  12. Basic properties of both sequences
  13. Even numbers with $c(n)=0$
  14. Odd numbers with $c(n)>0$
  15. Expectations
  16. ...and 1 more sections

Key Result

Theorem 1

The following statements hold.

Figures (15)

  • Figure 1: The unique inseparable odd-sized set with $n\leq 41$ points (left) and the smallest even-sized set not of type $[N,M,N,M]$ (right; see Proposition \ref{['prop-struc']}).
  • Figure 2: Two inseparable examples for $n=45$
  • Figure 3: Quartering of the set $\mathcal{P}$
  • Figure 4: Examples of all even $n$ with $c(n)=1$
  • Figure 5: Two inseparable examples for $n=18$ and the the smallest lacunary inseparable configuration with $8$-fold symmetry ($n=28$)
  • ...and 10 more figures

Theorems & Definitions (6)

  • Theorem
  • Definition
  • Lemma
  • Proposition 1
  • Proposition 2
  • Proposition 3