# Representations of zero as the sum of integers

Considering certain random walks I came up with the following question: Given a finite set $A$ containing positive and negative integers, how many representations of zero as the sum of $n$ integers in $A$ exist. In the case that $A$ has two elements the question is easy to answer but for set with larger cardinality the answer seems (to me) not obvious. Perhaps the question is to general, so I like to ask for which classes of sets $A$ there is closed formal or an asymptotics for the number of representations.

• How are you counting? For $A = \{1,-1\}$ and $n=2$ is the answer 1 or 2? Aug 24, 2016 at 22:30
• 2. Counting with order Aug 24, 2016 at 22:31
• So it seems that random walk can help. If $\xi_i$ are iid uniformly distributed in $A$ and $X_n = \xi_1 + \dots + \xi_n$ then your number should be $|A|^n P(X_n = 0)$ for which there should be lots of reasonable estimates. In particular you should expect qualitatively different answers depending on whether the walk is mean zero, i.e. whether $\sum A = 0$ or not. Aug 24, 2016 at 22:37