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Generating-Function Enumeration of Lattice Paths Weighted by Ascent Lengths

A generating-function framework for counting nonnegative lattice paths where each path contributes a weight given by a product over the lengths of its maximal blocks of consecutive up-steps. A master theorem yields a functional equation for the weighted generating function over broad families of step sets and weight functions, which Lagrange inversion then solves explicitly for generalized Catalan, Schroder, and Motzkin paths, connecting the counts to parking-function problems and OEIS sequences.

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Lattice paths enumerations weighted by ascent lengths Jun Yan∗ Abstract Recent work of the author

This paper studies weighted counting of lattice paths that stay on or above the x-axis, where each path is weighted by a product over the lengths of its maximal blocks of consecutive up-steps (its as…