TITLE: Motzkin polynomials and enumeration of chord diagrams
QUESTION [5 upvotes]: On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of the first few Motzkin polynomials listed in OEIS A055151. 
Can anyone provide a proof that this relation between the two sets of polynomials holds for all higher degrees in general?

REPLY [4 votes]: The bijection between planar partial chord diagrams and Motzkin paths is the following: the left end of a chord corresponds to U-step, the right end of a chord corresponds to D step, and "free" marked point corresponds to H step. For  example, the last diagram on Figure 3 in the paper is encoded by a UUHHDDUHHD Motzkin path. 
It is well-known, that the number of 2n-gon gluings into a sphere is n^th Catalan number, and so there is a bijection between planar polygon gluings and Dyck paths.
This is basically the same relation you mentioned.