Diagonal recurrence for noncrossing Chow polynomials

Find a recurrence that simultaneously extends alphabet size and word length along the diagonal (r,m)=(n,n+1), and determine whether this yields a useful scalar differential recurrence for the Chow polynomial H_{NC_{n+1}}(t).

Background

The proof of the paper’s main real-rootedness theorem uses a recurrence for Smirnov words with fixed alphabet size and varying word length. The noncrossing Chow polynomial corresponds instead to the diagonal specialization in which both the word length and alphabet size vary together.

The authors seek a recurrence adapted to this diagonal evolution. Such a recurrence could provide a direct comparison across n and potentially lead to a scalar differential recurrence, which might in turn support stronger structural results such as consecutive interlacing.