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Register a new userA refined enumeration of $p$-ary labeled trees
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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.
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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 vertex is called its indegree. Thus the local (resp. global) indegree sequence $lambda = 1^{e_1}2^{e_2} ldots$ of a tree on the vertex set $set{1,2,ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prufer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a $q$-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.
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 refinement $mathcal{O}_{n,k}$ of $mathcal{O}_n$, which is the set of ordered labeled trees whose maximal decreasing subtree has $k+1$ vertices.
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 that $b_m(n)$ can be represented as a $j$-fold summation by constructing a one-to-one correspondence between the $m$-ary partitions and a special class of integer sequences rely only on the base $m$ representation of $n$. It directly reduces to Andrews, Fraenkel and Sellers characterization of the values $b_{m}(mn)$ modulo $m$. Moreover, denote $c_{m}(n)$ the number of $m$-ary partitions of $n$ without gaps, wherein if $m^i$ is the largest part, then $m^k$ for each $0leq k<i$ also appears as a part. We also obtain an enumeration formula for $c_m(n)$ which leads to an alternative representation for the congruences of $c_m(mn)$ due to Andrews, Fraenkel, and Sellers.
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 $f$. We call it the minimal polynomial of $f$. It is in general a non-trivial task to find a concrete expression of the minimal polynomial of a given function, which has only been worked out for limited classes of functions in the literature. In this paper, we study minimal polynomial expressions of several functions that are closely related to some practically important procedures such as auction and voting.
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