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Let $G$ be a finite abelian group, and let $\eta(G)$ 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 $t\in [\exp(G)+1,\eta(G)-1]$ 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 $t\in [\exp(G)+1, \eta(G)-1]$ with the property above for some groups including the groups of rank two, and some special groups with large exponent., Comment: 19 pages

In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided., Comment: 6 pages

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 $\lambda$-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 $\lambda$-factorials of $n$ and establish the curious relations between the generating functions of general and exponential types for any sequence of numbers or polynomials., Comment: 13pages

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 $\sigma$ of $1,2,..., p$ such that $g=\sum_{i=1}^{p}(g_i\cdot h_{\sigma(i)})$. The question is related to a conjecture of A. Bialostocki concerning weighted subsequence sums and the Erd\H{o}s-Ginzburg-Ziv Theorem.