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 $k$-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 $Θ(D_n^k k<sup>β)$ in Theorem 3 where $D_n$ is the exponential growth constant for $n$ determined by the OGF and $β\geq n² - 1$.

• The paper presents a novel enumeration of DSCs via generating functions and Stirling number decompositions.
• It derives tight asymptotic bounds, revealing exponential growth modulated by polynomial factors linked to tree arity.
• It establishes Schur-convexity results to restrict colourings, informing algorithmic design in distributed systems.

Bounds on Decorated Sweep Covers in Tree Posets

Introduction and Problem Formulation

This paper investigates the enumeration and asymptotic analysis of decorated sweep covers (DSCs) in tree posets. A DSC is defined as a colouring of a maximal antichain in a tree-structured poset such that elements receiving the same colour are siblings, that is, they share a common parent in the tree. The DSC concept enables differentiation among parallel antichain elements that descend from a common ancestor, with immediate applications in algorithmic contexts requiring separation of structurally related parallel options, including distributed scheduling, multi-agent planning, and parallel Monte Carlo search.

The work focuses on the enumeration of $k$-coloured DSCs in infinite $n$-ary tree posets, seeking to characterize their combinatorial growth as a function of both the tree arity $n$ and the number of available colours $k$. The core technical challenge lies in managing the explosion of combinatorial possibilities introduced by the sibling-colouring constraint, especially as both $n$ and $k$ become large.

Structural Results and Generating Functions

The combinatorics of DSCs are encoded via a decomposition theorem, leveraging the observation that in a tree poset, every maximal antichain can be partitioned uniquely into sets of siblings. These sibling sets are then assigned colours under the restriction that each colour class appears only within a single sibling set. The enumeration thus naturally reduces to summing over Stirling numbers of the second kind for partitions of each sibling set, followed by a secondary summation over all valid distributions of the $k$ colours across sibling sets.

Key to the enumeration is a recursive construction that exploits the disjointness property present at each internal node of a tree poset. The associated recurrence leads to a closed form for the ordinary generating function (OGF) for the sequence $f_n(k)$ counting the number of $k$-coloured DSCs at each arity $n$: $n$0 where $n$1 is the $n$2-th Touchard polynomial, that is, $n$3. The OGF thus encodes the enumeration of DSCs for all $n$4 simultaneously and reveals subtle algebraic structure: for each fixed $n$5, $n$6 is an algebraic, D-finite power series.

Asymptotic Bounds and Schur Convexity

The authors derive tight asymptotic bounds on the exponential and polynomial growth rates of $n$7. The sequence exhibits strictly exponential growth with a polynomial modifier: $n$8 where $n$9 is an algebraically-specified exponential growth constant for each arity $n$0, and the exponent $n$1 satisfies $n$2.

A key technical component is the establishment of new Schur-convexity results for products of binomial coefficients over integer compositions. Explicit lower and upper bounds are derived, employing refined analytic combinatorics together with Stirling-type bounds. This allows for precise majorization and minorization of the contributions from sibling-set colour assignments, establishing that the DSC enumeration does not admit closed-form polynomial expressions in $n$3 for general arity, but is always bounded between sharp exponential and sub-exponential terms.

Analytical Discussion

A non-trivial implication of the main OGF is that, for each $n$4, the generating function is not solvable explicitly in radicals due to degree constraints (Abel-Ruffini theorem). Despite this, its algebraicity and D-finiteness guarantee both analytic tractability for singularity analysis and the effective computability of high-order terms. This property, when paired with the asymptotic exponential lower bound, fully characterizes the combinatorial explosion inherent in $n$5-DSCs on trees and points to the computational infeasibility of exact enumeration in the large-$n$6, large-$n$7 regime.

The sibling-colouring constraint structure—maximal antichains partitioned into siblings, and colour-class assignments restricted to siblings—yields a combinatorial landscape distinct from classical antichain or Sperner-type enumeration. The enumeration formulas and their asymptotics may find extended relevance in algorithmic structures relying on hierarchical decompositions, particularly in treewidth-based dynamic programming, hierarchical controller synthesis, and resource allocation over distributed systems with tree-shaped dependency graphs.

Theoretical and Practical Implications

The rigorous treatment of DSCs opens new directions for combinatorial analysis on poset-structured colourings with local constraints, beyond what is possible in classical antichain theory. Practically, the results clarify the parameter regimes in which algorithmic exploitation of DSCs is possible, and set theoretical limits on parallelism-injecting colourings for tree-like systems.

The explicit analytic bounds provide tools for asymptotic analysis of related constrained-colouring problems, and the recursive structure of the OGF enables the development of efficient symbolic computation and bootstrapped numerical algorithms for moderate $n$8. Furthermore, the established connection between the sibling-colouring constraint and Schur-convex products of binomials opens a pathway for generalization to broader classes of posets, including forests and multi-rooted trees.

From a theoretical standpoint, the paper advances the combinatorial toolkit for reasoning about structurally differentiated maximal antichains and provides analytic methodology that may extend to other locality-constrained partition or colouring problems, such as those arising in distributive lattice analysis or multi-agent task allocation.

Conclusion

The paper offers a comprehensive characterization of decorated sweep covers in tree posets, introducing explicit generating functions, establishing their algebraic properties, and proving sharp asymptotic bounds on their growth. The technical innovations around Stirling and Schur-convexity estimates contextualize the DSC enumeration among other poset-theoretic counting problems. These results establish both the computational hardness of the DSC counting problem in general scenarios and provide structured approaches and analytical guidance for algorithmic approximation and symbolic expansion in practical applications. The methodological approach and results are likely to invite further research into constraint-structured colourings in more general poset families and their applications in distributed and parallel computation.