AI 中文总结
研究有根无标号树生成的偏序集族中理想与反链的平均大小，利用生成函数和奇点分析证明二次根分解族中理想平均大小为 n/3，并分析反链常数及分支限制的影响。
AI 中文摘要
我们研究了由有根无标号树产生的若干偏序集族中理想与反链的平均大小。对于理想，我们通过将顶点染成红色来编码序理想，条件是红色顶点的每个后代也是红色的。利用生成函数和奇点分析，我们证明对于每个由二次根分解支配的族，大小为 $n$ 的树中红色顶点的平均数渐近于 $n/3$。这包括二元平面树、平面 1-2 树、平面 2 树和 0-1 树，而无限制的有根平面树的平均理想大小则渐近于 $2n/5$。对于反链，由两两不可比的蓝色顶点表示，相应的渐近常数取决于特定的分支函数；例如，二元平面树的平均反链大小渐近于 $n/6$。我们还考虑了每个顶点至多有 $k$ 个孩子的平面树，并证明红色顶点的渐近比例随 $k$ 严格增加，从 $k=2$ 时的 $1/3$ 到无限制有根平面树的极限值 $2/5$。证明基于双变量生成函数以及解析组合学的解析隐函数和平滑隐函数方法。
英文摘要
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.
Comments28 pages, 2 figures
