The Enumeration of Sequences with Restrictions on their Partial Sums
We examine sequences containing $p$ "$-t$"s and $pt+r$ "$+1$"s, where $p$, $t$, and $r$ are integers satisfying $p\ge0$, $t\ge 1$ and $pt+r\ge0$. We develop a rotation method to enumerate the number of sequences meeting additional requirements related to their partial sums. We also define downcrossings about $\ell$ and their downcrossing numbers, and obtain formulas for the number of sequences for which the sum of the downcrossing numbers equals $k$, for $\ell \le r+1$. We finish with an investigation of the first downcrossing number about $\ell$, for any $\ell$.