Generalized rank functions and quilts of alternating sign matrices

Billey, Sara C. (author), Konvalinka, Matjaž (author)

In this paper, we present new objects, quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on 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. Quilts 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.

• Language: English
• Keywords: quilts, alternating sign matrices, Dedekind numbers, distributive lattice
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.05.2026
• Year: 2026
• Number of pages: 45 str.
• Numbering: Vol. 491, article no. 110863
• PID: 20.500.12556/RUL-180858
• UDC: 519.1
• ISSN on article: 0001-8708
• DOI: 10.1016/j.aim.2026.110863
• COBISS.SI-ID: 272054531
• Publication date in RUL: 18.03.2026

Record is a part of a journal

• Title: Advances in mathematics
• Shortened title: Adv. math.
• Publisher: Elsevier
• ISSN: 0001-8708
• COBISS.SI-ID: 24891904

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.