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This paper deals with the enumeration of Grand-Dyck paths according to the statistic "number of occurrences of $\tau$" for every string $\tau$ of length $3$, taking into account the number of flaws of the path. Consequently, some new refinements of the Chung-Feller theorem are obtained.

Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property