Séminaire Lotharingien de Combinatoire, 93B.76 (2025), 12 pp.
Sara Billey and Matjaž Konvalinka
Quilts of Alternating Sign Matrices
Abstract.
In this extended abstract, we present new objects, quilts of
alternating sign matrices with respect to two given posets.
%%
For example, the rank function on all submatrices of a matrix gives rise
to a quilt with respect to two Boolean lattices.
%%
When the two posets are chains, a quilt is equivalent to an
alternating sign matrix and its corresponding corner sum matrix. Quilts
also generalize the monotone Boolean functions counted by the Dedekind
numbers. They form a distributive lattice with many beautiful
properties and contain many classical and well known sublattices, such
as the lattice of matroids of a given rank and ground set. While
enumerating quilts is hard in general, we prove two major enumerative
results, when one of the posets is an antichain and when one of them
is a chain. We also give some bounds for the number of quilts when one
poset is the Boolean lattice.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
The following versions are available: