Parking functions, Smirnov words, and noncrossing Chow polynomials

Source abstract

We prove real-rootedness of the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying a last-letter interlacing recurrence. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric $g$-contributions of Ehrenborg--Hetyei--Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q.~Xiao recently proved it independently by a different differential recurrence. We also give a separate finite Schur--Szegő convolution proof of the individual statement. These results prove Conjecture~4.2 of Xiao and Conjecture~11.2 of Ehrenborg--Hetyei--Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.