On the Average Sizes of Ideals and Antichains in Some Infinite Families of Posets

Abstract

We study the average sizes of ideals and antichains in several families of posets arising from rooted unlabeled trees. For ideals, we encode order ideals by coloring vertices red, with the condition that every descendant of a red vertex is also red. Using generating functions and singularity analysis, we show that for every family governed by a quadratic root decomposition, the average number of red vertices in a tree of size $n$ is asymptotic to $n/3$. This includes binary plane trees, plane 1-2 trees, plane 2-trees, and 0-1-trees, while unrestricted rooted plane trees instead have average ideal size asymptotic to $2n/5$. For antichains, represented by pairwise incomparable blue vertices, the corresponding asymptotic constants depend on the particular branching function; for example, binary plane trees have average antichain size asymptotic to $n/6$. We also consider plane trees in which every vertex has at most $k$ children and prove that the asymptotic proportion of red vertices increases strictly with $k$, from $1/3$ when $k=2$ to the limiting value $2/5$ for unrestricted rooted plane trees. The asymptotic proportion of blue vertices also increases strictly with $k$, tending to $3/10$. The proofs are based on bivariate generating functions together with the analytic implicit-function and smooth implicit-function methods of analytic combinatorics.