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Generalized rank functions and quilts of alternating sign matrices

Sara Billey, Matjaž Konvalinka

Abstract

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.

Generalized rank functions and quilts of alternating sign matrices

Abstract

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.

Paper Structure

This paper contains 19 sections, 26 theorems, 93 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Posets and lattices
  4. Alternating sign matrices and Fubini words
  5. Matroids and flag matroids
  6. Dedekind maps and quilts
  7. Motivation from the space of spanning line configurations
  8. Properties of the quilt lattice
  9. Enumeration of antichain quilts
  10. Enumeration of chain quilts
  11. Enumeration of Boolean quilts
  12. Final remarks
  13. Generalized Dedekind Numbers
  14. Boolean-Chain numbers
  15. Antichain-Boolean numbers
  16. ...and 4 more sections

Key Result

Theorem 1.1

The number of quilts of type $(P,C_n)$ for $n \geq \operatorname{rank} P$ is a polynomial of degree $b(P) = \sum_{x \in P} \operatorname{rank} x$. Furthermore, the leading coefficient is $\frac{|S(P)|}{b(P)!}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Hasse diagrams for $C_3$, $A_2(3)$ and $B_3$.
  • Figure 2: The Dedekind graph (the loops are not shown) and the restricted Dedekind graph of $C_3$.
  • Figure 3: A visual representation of an element in $\operatorname{Quilts}(B_3,C_2)$
  • Figure 4: Another visualization of quilts of types $(B_{3},C_{2})$ and $(B_{3},C_{5})$.
  • Figure 5: The Hasse diagram of $\operatorname{Quilts}(C_2,B_3)$.
  • ...and 4 more figures

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Example 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 61 more