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Let $mathcal{T}^{(p)}_n$ be the set of $p$-ary labeled trees on ${1,2,dots,n}$. A maximal decreasing subtree of an $p$-ary labeled tree is defined by the maximal $p$-ary subtree from the root with all edges being decreasing. In this paper, we study a new refinement $mathcal{T}^{(p)}_{n,k}$ of $mathcal{T}^{(p)}_n$, which is the set of $p$-ary labeled trees whose maximal decreasing subtree has $k$ vertices.
For a labeled tree on the vertex set $set{1,2,ldots,n}$, the local direction of each edge $(i,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a ve
In this paper we enumerate the cardinalities for the set of all vertices of outdegree $ge k$ at level $ge ell$ among all rooted ordered $d$-trees with $n$ edges. Our results unite and generalize several previous works in the literature.
Let $mathcal{O}_n$ be the set of ordered labeled trees on ${0,...,n}$. A maximal decreasing subtree of an ordered labeled tree is defined by the maximal ordered subtree from the root with all edges being decreasing. In this paper, we study a new refi
Let $mge 2$ be a fixed positive integer. Suppose that $m^j leq n< m^{j+1}$ is a positive integer for some $jge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this paper, we show t
Let $mathbb{F}_p$ be the finite field of prime order $p$. For any function $f colon mathbb{F}_p{}^n to mathbb{F}_p$, there exists a unique polynomial over $mathbb{F}_p$ having degree at most $p-1$ with respect to each variable which coincides with $f