Bijections for rhombic alternative tableaux

We generalize well-known bijections between alternative tableaux and permutations to bijections between rhombic alternative tableaux (RAT) and assemblées of permutations. We show how these various bijections are connected. As a consequence, we find a refined enumeration formula for RAT. One of our bijections carries many statistics from RAT to assemblées; notably, it sends the number of free cells to the number of crossings, which answers a question of Mandelshtam and Viennot. We also find an $r!$-to-$1$ map from marked Laguerre histories to assemblées, answering a question of Corteel and Nunge.

💡 Research Summary

This paper extends the well‑known bijections between alternative tableaux and permutations to a broader setting involving rhombic alternative tableaux (RAT) and assemblées of permutations. The authors first recall the classical bijections—zigzag, insertion, and fusion‑exchange maps—between alternative tableaux and permutations, and then develop analogous constructions for RAT, which arise as combinatorial models for the two‑species partially asymmetric simple exclusion process (PASEP).

A rhombic diagram Γ_w is built from a word w∈{0,1,2}^n by drawing a southeast border (south, southwest, west steps) and a northwest border (west, southwest, south steps) and then tiling the enclosed region with three tile types: squares, tall rhombi, and short rhombi. Filling this diagram with up‑arrows, left‑arrows, and empty cells under the usual “no arrow points to another” rule yields a RAT of size (n,r), where r counts the number of 1’s (light particles) in w. When r=0 the objects reduce to ordinary alternative tableaux.

The first major contribution is a structural decomposition of RAT into indecomposable components called packed RAT. By recursively extracting maximal strips and using a “flattening map” that embeds a RAT into a subclass of extended alternative tableaux, the authors obtain a concise combinatorial proof of the refined enumeration formula
Y_{n,r}(α,β,1)=\binom{n}{r}(α+β+r)^{,n-r},
which had previously been proved via algebraic means.

Next, the authors generalize the insertion map Φ_{AT}^I and the zigzag map Φ_{AT}^Z to RAT. The insertion map proceeds by reading the labels of columns in decreasing order, inserting the row label of the up‑arrow and then the row labels of left‑arrows, thereby producing a permutation in S_{n+r}. They prove that this map is a bijection and that free rows of the RAT correspond exactly to right‑to‑left minima of the resulting permutation, while free columns correspond to right‑to‑left maxima. The zigzag map is defined by following a path that turns at every arrow; its output permutation coincides with the insertion output after applying the Foata transformation. Consequently, the three classical bijections are shown to be essentially the same up to simple relabelings.

A central new bijection is then introduced, which sends the number of free cells of a RAT to the number of crossings in the associated assemblée. Assemblées are unordered collections of disjoint permutations whose union is {1,…,n}; they can be visualized as signed permutations with arc diagrams. By interpreting each block as a set of arcs, the authors define a crossing statistic that extends the usual permutation crossing number. They prove that under their bijection, each free cell of the RAT contributes exactly one crossing, thereby answering the open problem posed by Mandelshtam and Viennot.

The paper also addresses a question of Corteel and Nunge concerning a factorization of the generating function for marked Laguerre histories. Marked Laguerre histories are Dyck‑type paths equipped with marked steps that encode the ASEP partition function. The authors construct an r!‑to‑1 map from marked Laguerre histories to assemblées, again using the crossing/arc‑diagram viewpoint. This map explains combinatorially why the generating function contains the factor (1+q)(1+q+q^2)…(1+q+…+q^{r-1}).

Finally, the authors discuss how these bijections preserve a suite of statistics: free rows ↔ RL‑minima, free columns ↔ RL‑maxima, free cells ↔ crossings, and how the partition function Z_{n,r}(α,β,q) = (αβ)^{,n-r} Y_{n,r}(α−1,β−1,q) can be expressed in terms of mixed moments of Al‑Salam–Chihara polynomials, linking the combinatorics to orthogonal polynomial theory.

In summary, the paper provides a unified framework that connects rhombic alternative tableaux, assemblées, and marked Laguerre histories through explicit, statistic‑preserving bijections. It resolves several open combinatorial questions, supplies refined enumeration formulas, and opens avenues for further exploration of multivariate generating functions and their relationships with Koornwinder and related families of orthogonal polynomials.