Motzkin polynomials and enumeration of chord diagrams

Question

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?

Edit 3/30/2023:

A further refinement of the array on p. 12 of Alexeev et al., reading along the diagonal, is A350499, the coefficients of the inverse noncrossing partitions / inverse refined Narayana polynomials / inverse parking functions (etc.) $[N^{(-1)}]$.

For example, starting with the first summand of $k = 2$ and plucking off the diagonal summands gives

$qs^2 + 3qs + 2$.

Compare this with

$m_3 - 3m_2m_2 + 2 m_1^2 = N^{(-1)}_n$.

Another example, starting with $k =4$, the diagonal is

$qs^4 + (5q + 5q^2) + (15 q + 15 q^2) + 35qs + 14$

compared with

$m_5 - 5 m_2 m_3 - 5 m_4 m_1 + 15 m_2^2 m_1 + 15 m_3 m_1^2 - 35 m_2 m_1^3 + 14 m_1^5 = N^{(-1)}_5.$

Finally, the initial numbers for the next diagonal

$(1, 6, 9,21, 42, 7, 56, 84, ?, Catalan \; 42?)$

are a coarsening of the coefficients of $N^{(-1)}_6$,

$(1, 6, (6, 3), 21, 42, 7, 56, 84, 126, 42).$

The array in the paper is derived from random matrix integration methods, which seem inevitably to arrive at some basic equations of free probability exemplified by eqns. 33-36.

Answer 1

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.