Bounds on Decorated Sweep Covers in Tree Posets

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 $n$.