Occurrence of negative coefficients in Taylor remainder polynomials: The case of skew Dyck paths and HD-free Schröder paths

Eu et al. (2002) [7] introduced new Taylor expansions for the generating functions of the Catalan and Motzkin numbers, establishing the first connection between analytic expansions and enumerative combinatorics. A distinctive feature of such expansions is that the Taylor remainder term is expressed as a polynomial in the generating function itself. However, in all previously studied cases, the corresponding Taylor remainder polynomials contain only nonnegative coefficients, consistent with the nonnegativity of the underlying counting statistics. In contrast, this paper presents the first two combinatorial models whose associated remainder polynomials exhibit negative coefficients, namely skew Dyck paths and HD-free Schröder paths. As a further development, we reveal an equidistribution phenomenon between the two lattice path families with respect to the corresponding step-type statistics. Moreover, combinatorial interpretations for the coefficients of the corresponding Taylor remainder polynomials are presented.