[Paper Review] 2-enumerations of halved alternating sign matrices

Theresia Eisenkölbl|ArXiv.org|Jun 6, 2001

Graph theory and applications3 citations

TL;DR

This paper proves three conjectures by Jim Propp concerning 2-enumerations of halved alternating sign matrices (ASMs) with specific height matrix constraints. It establishes that weighted enumerations—using weights based on the number of −1s and 1s in even/odd positions—yield exact counts equal to the number of perfect matchings in halved Aztec diamonds and fortress graphs, respectively, with closed-form expressions involving powers of 2 and 5.

ABSTRACT

We compute 2-enumerations of certain halved alternating sign matrices. In one case the enumeration equals the number of perfect matchings of a halved Aztec diamond. In the other case the enumeration equals the number of perfect matchings of a halved fortress graph. Our results prove three conjectures by Jim Propp.

Motivation & Objective

• To resolve three conjectures by Jim Propp regarding weighted enumerations of halved alternating sign matrices with specific height matrix structures.
• To establish exact combinatorial formulas for 2-enumerations of halved ASMs under different weightings based on the positions of −1s and 1s.
• To connect the enumeration of these restricted halved ASMs to known combinatorial objects: perfect matchings in halved Aztec diamonds and fortress graphs.
• To provide a rigorous proof for the weight $ 2^{N_{-}(A)} $ yielding $ 2^{n^2} $, and for the more complex weight $ 2^{N_{-}(A,\text{even}) + N_{+}(A,\text{odd})} $ yielding $ 3^n 5^{inom{n}{2}} $.

Proposed method

• Uses the correspondence between halved alternating sign matrices and perfect matchings in halved Aztec diamonds, leveraging a known 1-to-2^{N_{-}(A)} bijection.
• Applies height matrix constraints to restrict the class of halved ASMs to those with specific boundary conditions and central column entries $ c_i = n \pm 1 $.
• Employs combinatorial enumeration techniques based on position-based weighting: distinguishing between even and odd positions of −1s and 1s in the matrix.
• Uses symmetry arguments to show equivalence under reflection: swapping even and odd positions preserves the form of the height matrix and the weight function.
• Relies on known results from the literature: the number of perfect matchings in an Aztec diamond of order $ n-1 $ is $ 2^{inom{n}{2}} $, and in a $ 2n \times 2n $ fortress graph is $ 5^{n^2} $.
• Applies recursive or structural decomposition techniques (implied by the proof sketch for Theorem 3) to handle the case with fixed $ c_i = n \pm 1 $, yielding different formulas for even and odd $ n $.

Experimental results

Research questions

• RQ1 What is the 2-enumeration of halved alternating sign matrices with height matrices of the form (1), weighted by $ 2^{N_{-}(A)} $?
• RQ2 What is the 2-enumeration of halved ASMs with the same height matrix form, but weighted by $ 2^{N_{-}(A,\text{even}) + N_{+}(A,\text{odd})} $?
• RQ3 What happens to the enumeration when the central column entries $ c_i $ are constrained to be $ n+1 $ or $ n-1 $, and how does the result depend on the parity of $ n $?
• RQ4 How do these weighted enumerations relate to perfect matchings in halved Aztec diamonds and fortress graphs?
• RQ5 Can the symmetry of the height matrix class under vertical reflection be used to justify weight invariance between even and odd positions?

Key findings

• The weighted enumeration of halved ASMs of order $ 2n $ with height matrix of the form (1) and weight $ 2^{N_{-}(A)} $ is exactly $ 2^{n^2} $, matching the number of perfect matchings in a halved Aztec diamond.
• The weighted enumeration with weight $ 2^{N_{-}(A,\text{even}) + N_{+}(A,\text{odd})} $ yields $ 3^n 5^{inom{n}{2}} $, a result that matches a known formula for a related class of ASMs.
• When $ c_i = n+1 $ for all $ i $, the enumeration is $ 5^{inom{n}{2}} $ for even $ n $, and $ 2^n 5^{inom{n}{2}} $ for odd $ n $.
• When $ c_i = n-1 $ for all $ i $, the enumeration is $ 2^n 5^{inom{n}{2}} $ for even $ n $, and $ 5^{inom{n}{2}} $ for odd $ n $, showing a parity-dependent behavior.
• The results confirm Propp’s conjectures regarding 2-enumerations of halved ASMs under specific height constraints and weight functions.
• The paper establishes a direct link between restricted halved ASMs and perfect matchings in combinatorial graphs, providing exact closed-form expressions for their counts.