A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the $q,t$-Catalan Symmetry

Abstract

Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\bigcup_{n\ge 0}D_n$, and let $area, dinv, bounce \colon D\to\mathbb{N}$ be the standard statistics. Then, \[ C_n(q,t)=\sum_{P\in D_n}q^{area(P)}t^{bounce(P)} =\sum_{P\in D_n}q^{dinv(P)}t^{area(P)}. \] Haglund's zeta map $\zeta\colon D\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $\zeta$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a \emph{rank sort} and introduce \emph{weighted-rank polyregular maps} (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $\zeta^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $\zeta$ cannot be extended within WRP to exchange the two statistics.