Rational Dyck Excursion Counts as Sums of Generalized Fuss-Catalan Products

Source abstract

For walks with jumps +p and -q, where p,q are coprime positive integers, we express the weak and strict rational Dyck excursion counts as finite weighted sums of generalized Fuss-Catalan products. Both sums run over the same partitions of q for every path size. Reversing the order of the steps and changing their signs transforms a returning walk with jumps +p and -q into one with jumps +q and -p, with the same heights in reverse order. This preserves both weak and strict excursions. Applying the formulas with p and q exchanged therefore gives sums over partitions of p, so we can choose the smaller of the two jump magnitudes. The indices of the Fuss-Catalan factors satisfy certain divisibility conditions that are determined by the jump magnitudes and the partition parts. The weak formula alternates in sign with the number of parts, whereas every term in the strict formula is positive. We prove both formulas by cutting unrestricted walks at their first and last visits to their lowest height. Expanding the resulting factors produces the partitions of q and the different signs. We also give equivalent binomial convolutions. Multiplying each Fuss-Catalan factor by q in the weak formula or p in the strict formula gives a positive integer. For q=1, the weak and strict counts reduce to Fuss-Catalan and shifted Raney numbers, respectively; for q=2 and q=3, the formulas yield explicit convolution identities.