Counting depth zero patterns in ballot paths (Q2882921)

In this paper ballot paths are considered to be strings on two letters \(u\) and \(r\), in which no prefix has more \(r\)'s than \(u\)'s. If in the ballot path the number of \(r\)'s equals the number of \(u\)'s the well-known Dyck path is obtained. Using finite operator calculus, recursive formulas are given for the number of ballot paths with a given number of occurrences of a pattern, and for patterns with depth zero, a closed formula is obtained, too. The results are also extended to the case of two given patterns.