Symmetric statistics on rational Dyck paths

Rational Dyck paths are the rational generalization of classical Dyck paths. They play an important role in Catalan combinatorics, and have multiple applications in algebra and geometry. Two statistics over rational Dyck paths called run and ratio-run are introduced. They both have symmetric joint distributions with the return statistic. We give combinatorial proofs and algebraic proofs of the symmetries, generalizing a result of Li and Lin.

💡 Research Summary

The paper investigates symmetric joint distributions of statistics on rational Dyck paths, which are lattice paths from (0,0) to (m,n) staying weakly above the line y = (n/m)x. The authors introduce two new statistics, run and ratio‑run (denoted g‑run), that generalize the classical “run” statistic from ordinary Dyck paths to the rational setting. The run statistic is defined as the smallest integer i such that i does not appear in the co‑area sequence U(P) of a path P; equivalently, it counts the number of peaks before the first occurrence of the pattern EE. The ratio‑run statistic uses the integer k = ⌊m/n⌋ and counts the smallest i for which k·i is missing from U(P), thereby measuring peaks in columns spaced by k before the first EE pattern.

The central results are two involutions that preserve the composition type of a path (the sequence of lengths of maximal consecutive north steps). The first involution Φ maps a path P to Φ(P) such that run(Φ(P)) = ret(P) and ret(Φ(P)) = run(P), where ret(P) is the number of north steps that lie on the boundary of diagonal cells (the “return” statistic). The second involution Ψ satisfies g‑run(Ψ(P)) = ret(P) and ret(Ψ(P)) = g‑run(P). Consequently, the pairs (run, ret) and (g‑run, ret) each have symmetric joint distributions: for any non‑negative integers a, b, the number of paths with run = a and ret = b equals the number with run = b and ret = a, and similarly for g‑run.

The proofs are combinatorial and rely on a “hit and lift” algorithm. Each path is decomposed into vertical components M_i = N^{α_i}E, where α_i are the parts of the composition comp(P). The authors identify a minimal index r = rr(α,m) (depending only on the composition and the rectangle size) such that the first r components simultaneously contribute to both run and return. The remaining east steps are encoded as a binary signature sequence (0 for an east step not preceded by a north step, 1 otherwise). By systematically “lifting” EN pairs—replacing a pattern EN with a north step followed later by an east step—the algorithm rearranges the signature while keeping the composition unchanged. This operation swaps the roles of run and return (or g‑run and return) and yields the desired involutions.

Beyond the bijective proofs, the authors provide generating‑function identities. In the classical case (m = n) they recover known q‑t Catalan generating functions, and for the Fuss–Catalan case (m = kn) they obtain analogous formulas involving k‑Catalan numbers. These identities give an algebraic proof of the symmetry, complementing the combinatorial construction.

The paper also discusses connections to Shi hyperplane arrangements, parking functions, and Kupisch series of Nakayama algebras. The return statistic corresponds to the number of bounded regions in the Shi arrangement and to the repeated projective dimension in the Kupisch series. The run statistic, originally introduced for parking functions, encodes chamber inequalities. By extending these statistics to rational Dyck paths, the authors suggest new bridges between lattice‑path combinatorics and geometric representation theory.

Finally, several open problems are posed: (1) exploring multivariate symmetries involving run, g‑run, and ret simultaneously; (2) relating the new statistics to other well‑studied statistics such as bounce, area, dinv, and ddinv; (3) extending the framework to non‑integral slopes (i.e., rational slopes with denominator not dividing the numerator) and investigating whether analogous involutions exist.

In summary, the work enriches the theory of rational Dyck paths by introducing run‑type statistics, establishing their symmetric joint distributions with the return statistic via explicit composition‑preserving involutions, and linking these combinatorial phenomena to algebraic structures in hyperplane arrangements and representation theory. The results open new avenues for research on symmetry phenomena in Catalan‑type combinatorics.