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Jujuan Zhuang

Publications and source records attributed to Jujuan Zhuang.

4 recordsLinked to original sources
arXiv2011

On short zero-sum subsequences of zero-sum sequences

Let $G$ be a finite abelian group, and let $η(G)$ be the smallest integer $d$ such that every sequence over $G$ of length at least $d$ contains a zero-sum subsequence $T$ with length $|T|\in [1,\exp(G)]$. In this paper, we investigate the question whether all non-cyclic finite abelian groups $G$ share with the following property: There exists at least one integer $t\in [\exp(G)+1,η(G)-1]$ such that every zero-sum sequence of length exactly $t$ contains a zero-sum subsequence of length in $[1,\exp(G)]$. Previous results showed that the groups $C_n^2$ ($n\geq 3$) and $C_3^3$ have the property above. In this paper we show that more groups including the groups $C_m\oplus C_n$ with $3\leq m\mid n$, $C_{3^a5^b}^3$, $C_{3\times 2^a}^3$, $C_{3^a}^4$ and $C_{2^b}^r$ ($b\geq 2$) have this property. We also determine all $t\in [\exp(G)+1, η(G)-1]$ with the property above for some groups including the groups of rank two, and some special groups with large exponent.

math.NT↗
arXiv2010

$λ$-factorials of $n$

Recently, by the Riordan's identity related to tree enumerations, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, \end{eqnarray*} Sun and Xu derived another analogous one, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, \end{eqnarray*} where $D_{k}$ is the number of permutations with no fixed points on $\{1,2,\dots, k\}$. In the paper, we utilize the $λ$-factorials of $n$, defined by Eriksen, Freij and W$\ddot{a}$stlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and another two algebraic proofs. Using the umbral representation of our generalized identity and the Abel's binomial formula, we deduce several properties for $λ$-factorials of $n$ and establish the curious relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.

math.CO↗
arXiv2007

Weighted Sequences in Finite Cyclic Groups

Let $p>7$ be a prime, let $G=\Z/p\Z$, and let $S_1=\prod_{i=1}^p g_i$ and $S_2=\prod_{i=1}^p h_i$ be two sequences with terms from $G$. Suppose that the maximum multiplicity of a term from either $S_1$ or $S_2$ is at most $\frac{2p+1}{5}$. Then we show that, for each $g\in G$, there exists a permutation $σ$ of $1,2,..., p$ such that $g=\sum_{i=1}^{p}(g_i\cdot h_{σ(i)})$. The question is related to a conjecture of A. Bialostocki concerning weighted subsequence sums and the Erdős-Ginzburg-Ziv Theorem.

math.CO↗