发表机构
Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从自动机理论视角，证明见证q,t-卡特兰对称性的ζ映射及高度扫描双射非多正则映射，引入WRP映射后，发现ζ的逆映射不在WRP中，且WRP无法实现保持半长度的面积与Dinv交换，揭示了相关计算障碍。
AI 中文摘要
代数组合学常寻求双射以解释分布之间的逐对象恒等式。将组合对象编码为字，可使自动机理论将此类双射研究为字到字的计算，并测量其内存、输入访问及输出顺序控制能力，这通过询问双射所需的计算机制，细化了存在性问题。我们将此视角应用于Dyck路径，其研究动机为q,t-卡特兰多项式：设Dₙ为半长度n的Dyck路径集合，D=∪ₙ≥₀Dₙ，area、dinv、bounce∶D→ℕ为标准统计量，则Cₙ(q,t)=∑ₚ∈Dₙq^(area(P))t^(bounce(P))=∑ₚ∈Dₙq^(dinv(P))t^(area(P))。Haglund的ζ映射ζ∶D→D给出了双射证明：它保持半长度，且将(dinv,area)映射为(area,bounce)；相比之下，完全对称性Cₙ(q,t)=Cₙ(t,q)仍缺乏直接解释，目前尚无明确、统一、保持半长度的双射可在每条Dyck路径上交换area与dinv。自动机理论中的多正则映射提供了自然的计算起点，但我们证明ζ映射及见证Narayana对称性的经典高度扫描双射均非多正则映射；缺失的机制是由数值层级的全局排序，其范围随输入增长，我们称之为“秩排序”，并引入加权秩多正则映射（WRP），它通过此类排序扩展了多正则映射，且包含上述两种双射，不过WRP是确定性对数空间的真子类。我们证明ζ⁻¹映射不属于WRP，且无WRP映射可实现保持半长度的area-dinv交换，因此ζ映射背后的秩排序策略无法在WRP框架内扩展以交换这两个统计量。
英文摘要
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 $ζ\colon D\to D$ 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 $ζ$ 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 $ζ^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $ζ$ cannot be extended within WRP to exchange the two statistics.
Comments76 pages, Lean formalisation available at https://github.com/hongseok-yang/automata-catalan-symmetry-release
