  This article introduces and investigates a refinement of alternating sign
trapezoids by means of Catalan objects and Motzkin paths. Alternating sign
trapezoids are a generalisation of alternating sign triangles, which were
recently introduced by Ayyer, Behrend and Fischer. We show that the number of
alternating sign trapezoids associated with a Catalan object (resp. Motzkin
path) is a polynomial function in the length of the shorter base of the
trapezoid. We also study the rational roots of the polynomials and formulate
several conjectures and derive some partial results. Lastly, we deduce a
constant term identity for the refined counting of alternating sign trapezoids.
