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Signotopes with few plus signs

Helena Bergold, Lukas Egeling, Hung. P. Hoang

TL;DR

This work analyzes signotopes, a structured subclass of pseudohyperplane arrangements, through the lens of the higher Bruhat order. It proves that for a fixed difference $d=n-r$, the lower (and symmetrically the upper) levels of the order have the same size across all $(n,r)$ with that difference, via a bijection that preserves the partial order. The authors reinterpret signotopes as co-signotopes and leverage a Ferrers diagram–like local structure of one-element extensions of cyclic arrangements to construct the bijection, decomposing objects into independent components. They also develop a Ferrers-diagram–based enumeration of the plus-count, providing exact formulas for small parameters and connecting to classical partition functions, with tightness and conjectures guiding future work. Overall, the paper advances understanding of signotope enumeration under fixed-difference constraints and provides a robust combinatorial framework for componentwise analysis and counting.

Abstract

Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and Schechtman (1989), the higher Bruhat order is a natural order of $r$-signotopes on $n$ elements, with the signotope corresponding to the cyclic arrangement as the minimal element. In this paper, we show that the lower (and by symmetry upper) levels of this higher Bruhat order contain the same number of elements for a fixed difference $n-r$. This result implies that given the difference $d=n-r$ and $p$, the number of one-element extensions of the cyclic arrangement of $n$ hyperplanes in $\mathbb{R}^d$ with at most $p$ points on one side of the extending pseudohyperplane does not depend on $n$, as long as $n \geq d + p$.

Signotopes with few plus signs

TL;DR

This work analyzes signotopes, a structured subclass of pseudohyperplane arrangements, through the lens of the higher Bruhat order. It proves that for a fixed difference , the lower (and symmetrically the upper) levels of the order have the same size across all with that difference, via a bijection that preserves the partial order. The authors reinterpret signotopes as co-signotopes and leverage a Ferrers diagram–like local structure of one-element extensions of cyclic arrangements to construct the bijection, decomposing objects into independent components. They also develop a Ferrers-diagram–based enumeration of the plus-count, providing exact formulas for small parameters and connecting to classical partition functions, with tightness and conjectures guiding future work. Overall, the paper advances understanding of signotope enumeration under fixed-difference constraints and provides a robust combinatorial framework for componentwise analysis and counting.

Abstract

Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and Schechtman (1989), the higher Bruhat order is a natural order of -signotopes on elements, with the signotope corresponding to the cyclic arrangement as the minimal element. In this paper, we show that the lower (and by symmetry upper) levels of this higher Bruhat order contain the same number of elements for a fixed difference . This result implies that given the difference and , the number of one-element extensions of the cyclic arrangement of hyperplanes in with at most points on one side of the extending pseudohyperplane does not depend on , as long as .

Paper Structure

This paper contains 11 sections, 17 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof overview for Theorem \ref{['thm:bijection']}
  3. Co-signotopes
  4. Proof strategy of Theorem \ref{['thm:bijection']}
  5. Structure of $+$-subsets of a co-signotope in $\bar{\mathcal{S}}_p(n,d)$
  6. Co-signotopes with only one $+$-component
  7. The $+$-component decomposition of a co-signotope
  8. Proof of Theorem \ref{['thm:bijection']}
  9. Plus count
  10. Concluding remarks
  11. The counts of $S_{\leq 2}(n,1)$ and $S_{\leq 2}(n+1,2)$

Key Result

Theorem 1

Let $n, \tilde{n}, r, \tilde{r},d, p$ be natural numbers such that $n-r = \tilde{n}-\tilde{r} = d$ and $p \leq \min\{r,\tilde{r}\}$. Then there exists a bijection between $\mathcal{S}_{\leq p}(n,r)$ and $\mathcal{S}_{\leq p}(\tilde{n},\tilde{r})$ that preserves the relation $\preceq$.

Figures (2)

  • Figure 1: Two examples of one-element extensions of the cyclic arrangements $\mathbf{X}^{5,2}_c$ (left) and $\mathbf{X}^{6,2}_c$ (right). The red dashed lines are the extending pseudolines. To avoid clusters, we label each point of the arrangement by the lines that contain it (e.g., point 12 is the intersection of lines $H_1$ and $H_2$). The positive sides are the sides above the extending pseudolines. The corresponding two signotopes are mapped to each other in our bijection for \ref{['thm:bijection']}.
  • Figure 2: Left: A Ferrers diagram with the corresponding integer partition of 15, where the bottom-leftmost point corresponds to (1,1). Note that this is also a $(2,1)$-Ferrers diagram with respect to 15. Right: A $(2,2)$-Ferrers diagram with respect to 11. Since $a_2 \geq a_1$, there cannot be any point below the dashed line. The numbers are the count of points in each vertical line. Observe that these numbers must be strictly decreasing.

Theorems & Definitions (36)

  • Definition 1: Signotope
  • Definition 2: Higher Bruhat Order
  • Definition 3: Cyclic arrangement
  • Theorem 1
  • Corollary 4
  • Theorem \ref{thm:bijection}: Rephrased
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 26 more