Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity

TL;DR

The paper investigates vertically symmetric alternating sign matrices ($2n+1$-order) and their relationship to holey, symmetric lozenge tilings, extending both objects with an $n+3$ parameter refinement and proving that their multivariate generating functions coincide. By working through arrowed monotone triangles and a sequence of determinant/Jacobi–Trudi-type reformulations, the authors derive signless lattice-path descriptions (via LGV) and establish a combinatorial bridge to plane partitions, symplectic Schur polynomials, and totally symmetric self-complementary plane partitions, echoing Cauchy-type identities. The work yields an equinumeracy result as a corollary, and provides multiple path-model interpretations (including two additional families) that support a conjectured bijective correspondence akin to RS-K, while posing open questions about multivariate weights on both sides. Overall, the results deepen the connections among ASMs, VSASMs, lozenge tilings, and symmetric plane partitions, and suggest new bijective avenues inspired by classical Cauchy and RS-K theory.

Abstract

Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size $2$ that exhibit a cyclical as well as a vertical symmetry, but finding an explicit bijection proving this belongs to the most difficult problems in bijective combinatorics. Towards constructing such a bijection, we generalize the result by introducing certain natural extensions for both objects along with $n+3$ parameters and show that the multivariate generating functions with respect to these parameters coincide. This is a significant step from a constant number of equidistributed statistics to a linear number of statistics in $n$. The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects and then applying certain sign-reversing involutions. Another main result concerns the expansion of the multivariate generating function into symplectic characters as a sum over totally symmetric self-complementary plane partitions, which is in perfect analogy to the situation for ordinary ASMs where the Schur expansion can be written as a sum over totally symmetric plane partitions. This is exciting as it is reminiscent of the well-known Cauchy identity, and the Cauchy identity does have a bijective proof using the Robinson-Schensted-Knuth correspondence, and thus the result raises the question of whether there is a variation of the Robinson-Schensted-Knuth correspondence that does eventually lead to a bijective proof.

Figures (15)

• Figure 1: Cyclically and vertically symmetric lozenge tiling of a hexagon with a central triangular hole and the corresponding family of nonintersecting lattice paths. The gray tilings are forced due to the symmetry.
• Figure 2: Example of the lattice paths in Theorem \ref{['interpret1']} for $n=6$. The associated permutation is $\sigma = (1\;2\;3\;6\;4\;5)$ and the weight is $(-u v)^5 w^9 X_1^5 X_2^5 X_3^5 X_4^5 X_5^5 X_6^5(u X_3+v X_3^{-1}) (u X_6 + v X_6^{-1})$. In the second region, we draw the even and odd paths in different colors.
• Figure 3: An example of the bijective correspondence between cyclically symmetric lozenge tilings of a holey hexagon and descending plane partitions. The lozenge tiling on the left side is the same as in Figure \ref{['fig:ExampleRhombusTiling']} but rotated by $30^\circ$. The dotted lines mark a third of the lozenge tiling as the fundamental area.
• Figure 4: Example of a lattice path interpretation of \ref{['LGV1']} with $i=5$, $j=3$, $p=8$ and $q=6$. The steps contribute the factor $-u v w^2 (u X_1+v X_1^{-1})^2 (u X_3 + v X_3^{-1})$ to the weight of the path. In the second region, the path is even.
• Figure 5: Fundamental area $\mathcal{H}_n$

Theorems & Definitions (29)

• Definition 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 4.1
• Lemma 4.2