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Bounds on Decorated Sweep Covers in Tree Posets

Blake A. Wilson, Colin Krawchuk

Abstract

We introduce decorated sweep covers as a colouring on maximal antichains in tree posets such that if two elements have the same colour they are siblings. DSCs appear in applications wherever maximal antichains require structural differentiation among parallel options that have a common ancestry, e.g., distributed systems, drone routing in logistics, and Monte Carlo Tree Search. We restrict our analysis to enumerating $k$-coloured DSCs in $n$-ary tree posets and prove i) their ordinary generating function in Theorem 1, ii) new Schur-convexity results for binomial coefficients in Theorem 2 and iii) bounds on the OGF coefficients which scale as $Θ(D_n^k k^β)$ in Theorem 3 where $D_n$ is the exponential growth constant for $n$ determined by the OGF and $β\geq n^2 - 1$.

Bounds on Decorated Sweep Covers in Tree Posets

Abstract

We introduce decorated sweep covers as a colouring on maximal antichains in tree posets such that if two elements have the same colour they are siblings. DSCs appear in applications wherever maximal antichains require structural differentiation among parallel options that have a common ancestry, e.g., distributed systems, drone routing in logistics, and Monte Carlo Tree Search. We restrict our analysis to enumerating -coloured DSCs in -ary tree posets and prove i) their ordinary generating function in Theorem 1, ii) new Schur-convexity results for binomial coefficients in Theorem 2 and iii) bounds on the OGF coefficients which scale as in Theorem 3 where is the exponential growth constant for determined by the OGF and .

Paper Structure

This paper contains 10 sections, 17 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Methods
  4. OGF for DSCs in Infinite $n$-ary Trees
  5. Proof of Thm. \ref{['thm:ogf']}
  6. OGF Coefficient Bounds via Bootstrapping
  7. Proof of Thm. \ref{['thm:bounds']}
  8. Appendix
  9. Bounds on Compositions
  10. Proof of Theorem \ref{['thm:bin']}

Key Result

Theorem 1

The number of lower $k$-DSCs in an infinite, $n$-ary tree poset is generated by the ordinary generating function (OGF) where $B_n(z) = \sum_{k=0}^n \genfrac\{\}{0pt}{}{n}{k} z^k$ is the $n$-th Touchard polynomial. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A 2‐level tree poset: on the left the maximal antichain $S = \{p_{11}, p_{21}, p_{22}\}$ (bottom level) is shaded grey; on the right the same maximal antichain is coloured as a DSC $c(p_{11}) = \text{red}$, $c(p_{21}) = c(p_{22}) = \text{blue}$.

Theorems & Definitions (34)

  • Theorem 1: Simplified
  • Theorem 2
  • Theorem 3
  • Definition 1: Decorated Sweep Cover (DSC)
  • Corollary 1
  • Lemma 1: Enumeration via Disjointness
  • proof
  • Corollary 2
  • proof
  • Lemma 2: DSC Decomposition
  • ...and 24 more