Bounds on Decorated Sweep Covers in Tree Posets

TL;DR

This paper studies decorated sweep covers in tree posets, providing generating functions, Schur-convexity results, and bounds on their enumeration, with applications in distributed systems and logistics.

Contribution

It introduces decorated sweep covers, derives their generating functions, proves new Schur-convexity results, and establishes bounds on their enumeration in tree posets.

Findings

Generated explicit formulas for decorated sweep covers in tree posets.

Proved Schur-convexity properties for binomial coefficients related to the problem.

Established bounds on the number of decorated sweep covers scaling as _n^k k^eta.

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 $\Theta(D_n^k k^\beta)$ in Theorem 3 where $D_n > n$ is the exponential growth constant for each $n$.