On the area-depth symmetry on Łukasiewicz paths

In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\mathtt{depth}$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $\mathtt{area}$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $\mathtt{depth}$ to so-called ``$\vec{k}$-Dyck paths’’. They showed that $\mathtt{area}$ and $\mathtt{depth}$ are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of Łukasiewicz paths by interpreting $\mathtt{area}$ and $\mathtt{depth}$ under the classical bijection between Łukasiewicz paths and plane trees, through which the symmetry is transparent.

💡 Research Summary

The paper addresses a conjecture posed by Qu and Zhang (2025) concerning the joint distribution of two statistics—area and depth—on Łukasiewicz paths when both the first and the last up‑steps are fixed. The authors work in the broader setting of Łukasiewicz paths, which generalize the k‑Dyck paths studied earlier, and they exploit the classical bijection between Łukasiewicz paths and rooted plane trees.

After recalling the definition of Łukasiewicz paths (steps (1,k) with k ≥ −1, ending with a down‑step) they introduce two statistics. The area of a path is the sum of the y‑coordinates of its up‑steps. Depth is defined via a matching between each down‑step (except the final one) and the nearest preceding up‑step that still needs down‑steps; a recursive labeling then yields the depth vector, whose sum is the depth of the path. This definition coincides with the “Filling” and “Ranking” tableaux used by Qu and Zhang.

The core of the argument is the bijection λ: T → P that maps a plane tree to a Łukasiewicz path by a contour walk (leaves become D, an internal node of degree k+1 becomes Uₖ) and its inverse τ. Under this correspondence, each up‑step corresponds to an internal node. The authors define, for an internal node u, the numbers lthorn(u) and rthorn(u) counting left‑ and right‑thorns—edges that descend from ancestors to the left or right of the path from the ancestor to u. Summing over all internal nodes yields lthorn(T) and rthorn(T).

Proposition 3.3 proves that the area of a path equals rthorn of the associated tree; Proposition 3.4 shows that depth equals lthorn. Both proofs proceed by induction on the distance from the root, carefully tracking how the y‑coordinate changes when moving from a parent node to a child and how the matching of down‑steps translates into thorn counts.

With these identifications, symmetry becomes a matter of tree reflection. Proposition 4.1 observes that reflecting a tree left‑right (mir) swaps lthorn and rthorn. Consequently, the composition λ ∘ mir ∘ τ is an involution on the set Lₐ, M of Łukasiewicz paths with a fixed first up‑step a and a fixed multiset M of the remaining up‑step degrees, and it exchanges area and depth. This yields Theorem 4.2, i.e. eCₐ, M(q,t)=eCₐ, M(t,q).

When the last up‑step is also fixed (value b), a simple reflection is insufficient because it would move the last up‑step. The authors therefore introduce a “lodestar swap”: identify the leftmost and rightmost internal nodes whose children are all leaves (the left and right lodestars) and interchange them. Proposition 4.4 shows that this swap does not affect lthorn or rthorn. Combining mir with the lodestar swap gives an involution λ ∘ swap ∘ mir ∘ τ that preserves both the first and last up‑step while swapping area and depth. This proves the main result, Theorem 1.1, confirming the conjecture that eCₐ, M, b(q,t) is symmetric in q and t.

Corollary 4.5 extends the symmetry to the case where only the last up‑step is fixed. Proposition 4.6 shows further consequences: (1) fixing the first up‑step does not affect the generating function when the multiset of other up‑steps is unchanged; (2) for profiles of length three the full q,t‑symmetry holds.

Finally, Remark 4.7 presents a refined generating function F(z,q,t; p₀,p₁,…) that records area and depth together with the multiset of up‑step degrees. Using the tree decomposition at the root and the symbolic method, the authors derive a recursive functional equation (3) for F, illustrating how the combinatorial structure translates into analytic form.

Overall, the paper provides a clean, bijective proof of the area‑depth symmetry for Łukasiewicz paths under various fixing conditions, thereby settling the conjecture of Qu and Zhang and offering a transparent combinatorial perspective on q,t‑symmetry in generalized Catalan families.