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arxiv: 2605.02668 · v1 · submitted 2026-05-04 · 🧮 math.CO

A generalization in affine type A of Coxeter sortable elements and Reading's bijection with noncrossing partitions

Jad Abou-Yassin This is my paper

Pith reviewed 2026-05-08 18:35 UTC · model grok-4.3

classification 🧮 math.CO
keywords affine symmetric groupc-sortable elementsnoncrossing partitionsbiclosed setspattern avoidancetranslation-invariant total ordersCoxeter groupsarc diagrams
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The pith

In the affine symmetric group, c-sortable elements are those whose one-line notations avoid certain patterns, with a bijection to c-noncrossing partitions via biclosed sets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper generalizes the notion of c-sortable elements and Reading's bijection with noncrossing partitions from finite Coxeter groups to the affine symmetric group. It does so by characterizing the sortable affine permutations through pattern avoidance in their one-line notation and by introducing c-sortable biclosed sets of reflections whose translation-invariant total orders avoid the same patterns. The bijection is built by chaining correspondences among biclosed sets, TITOs, c-noncrossing partitions of an annulus, and a subset of cyclic noncrossing arc diagrams. A sympathetic reader would care because the result supplies explicit combinatorial tests and equivalences that work in an infinite setting where length functions and inversion sets are less immediately tractable.

Core claim

The paper claims that an affine permutation is c-sortable precisely when its one-line notation avoids a specific collection of patterns. It defines c-sortable biclosed sets as those biclosed sets of reflections whose associated translation-invariant total orders on the integers avoid the same patterns. It then proves that these c-sortable biclosed sets stand in bijection with the c-noncrossing partitions by composing three known or constructed correspondences: biclosed sets with TITOs, c-noncrossing partitions with partitions of an annulus, and a distinguished subset of cyclic noncrossing arc diagrams with both collections.

What carries the argument

The translation-invariant total order (TITO) on the integers attached to each biclosed set of reflections, which coincides with the one-line notation of affine permutations and makes pattern avoidance a well-defined test for sortability.

If this is right

  • The pattern-avoidance test supplies a direct, length-independent way to decide whether a given affine permutation is c-sortable.
  • Enumerative or structural results about c-noncrossing partitions transfer immediately to statements about c-sortable biclosed sets.
  • A natural subset of cyclic noncrossing arc diagrams is placed in bijection with both the sortable biclosed sets and the noncrossing partitions.
  • The classical finite-type bijection of Reading is recovered as the restriction of the affine construction to the finite symmetric group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analogous pattern-avoidance characterizations may exist for sortable elements in other affine Coxeter groups once suitable total orders on their reflections are identified.
  • The c-sortable biclosed sets might admit a Cambrian-like lattice structure whose cover relations are visible in the arc-diagram or TITO representation.
  • Counting the pattern-avoiding TITOs could yield new generating functions for affine noncrossing partitions.

Load-bearing premise

The biclosed sets of reflections in the affine symmetric group admit a translation-invariant total order that matches the usual one-line notation and on which pattern avoidance exactly isolates the c-sortable elements.

What would settle it

An explicit affine permutation whose one-line notation avoids the stated patterns yet fails to be c-sortable, or a pair of c-sortable biclosed sets and c-noncrossing partitions whose images under the constructed maps are not in one-to-one correspondence.

Figures

Figures reproduced from arXiv: 2605.02668 by Jad Abou-Yassin.

Figure 1
Figure 1. Figure 1: Main objects and maps used in this paper alongside a moving example with view at source ↗
Figure 2
Figure 2. Figure 2: A c-noncrossing partition of an annulus for n = 10 i. Notation In this paper, we use a slightly different terminology compared to [BR24]: we call the em￾bedded blocks curved polygons for similarities with the case of the finite type A, and we omit the adjective « nondangling » for annular blocks as we only consider such annular blocks. ii. c-marking of an annulus Let c be a Coxeter element of the affine sy… view at source ↗
Figure 3
Figure 3. Figure 3: Two representations of the same cyclic noncrossing arc diagram on view at source ↗
Figure 4
Figure 4. Figure 4: Generalized rank two parabolic subgroups in the positive root system of type view at source ↗
Figure 5
Figure 5. Figure 5: Bijection between c-noncrossing partitions and c-noncrossing arc diagrams in type A We do a similar construction for cyclic c-noncrossing arc diagrams and c-noncrossing partitions. Using the definitions from Paragraph F of Section 1, for each polygon P of a c-noncrossing partition of an annulus and for 26 view at source ↗
Figure 6
Figure 6. Figure 6: Bijection between c-noncrossing partitions of an annulus and cyclic c-noncrossing arc diagrams The noncrossing property of the arc diagram obtained from this procedure comes from the fact that two curved polygons cross if and only if there exists a diagonal of each one such that these diagonals cross, where a diagonal is any segment between two vertices of a polygon. Proposition 3.14. The previous construc… view at source ↗
Figure 7
Figure 7. Figure 7: Example of the steps for renumbering a cyclic view at source ↗
Figure 8
Figure 8. Figure 8: Renumbering AL and AR 11 12 13 14 15 16 7 18 9 10 view at source ↗
Figure 9
Figure 9. Figure 9: The 2-numbering of the cyclic c-noncrossing arc diagram of Example 3.19 Proof. We need to show that for every class of chains or loops modulo n, there is exactly one representative in Na(A). In the case where Ca(A) is finite, of last element b ∈ Z, then all the integers b+pn for p ∈ Z have no neighbor arc and they are the final points of the chains they belong to. This means for each arc u → v in A, if eit… view at source ↗
Figure 10
Figure 10. Figure 10: The −1-numbering of B Remarks and questions Type Ce Under the classical folding of type Ae2n−3 into type Cen−1, the objects used in this paper are well behaved (see [BS24b, BR24]) and our results naturally restrict to this type. The argument is beyond the scope of this paper and will appear in the author’s PhD thesis. Other classical affine types Most combinatorial objects involved in this paper have gene… view at source ↗
read the original abstract

This paper generalizes in the affine symmetric group the notion of Coxeter sortable (or c-sortable for short) elements, as well as the classical bijection between c-sortable elements and c-noncrossing partitions defined by Reading in finite Coxeter groups. The generalization to the affine symmetric group of the c-sortable elements is achieved by using biclosed sets of reflections. Using recent works from Barkley and Speyer, these biclosed sets admit a sort of "one-line notation" called a TITO on $\mathbb{Z}$ (translation-invariant total order on $\mathbb{Z}$) that coincides with the usual one-line notation in the case of an affine permutation. We characterize the c-sortable elements of the affine symmetric group by pattern avoidance on their one-line notation, mirroring the well-known characterizations of c-sortable elements in the classical finite types. Based on this criterion, we then define the c-sortable biclosed sets, generalizing the c-sortable elements, as biclosed sets such that their TITOs on $\mathbb{Z}$ avoid certain patterns. We also build a bijection from our set of c-sortable biclosed sets to the set of c-noncrossing partitions using various combinatorial objects and their one-to-one correspondences. First, the TITOs, in bijection with the biclosed sets of the affine symmetric group using results from Barkley and Speyer. Second, the c-noncrossing partitions of an annulus, in bijection with the c-noncrossing partitions of the affine symmetric group, using results from Digne and Reading. Finally, the cyclic noncrossing arc diagrams, defined by Barkley, for which we exhibit a subset in bijection with both the set of c-sortable biclosed sets and the set of c-noncrossing partitions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper generalizes c-sortable elements of the affine symmetric group via biclosed sets of reflections. It first characterizes the c-sortable affine permutations by pattern avoidance in one-line notation (equivalently, in their associated TITOs), then defines c-sortable biclosed sets as those whose TITOs avoid the same patterns. A bijection is constructed from these c-sortable biclosed sets to c-noncrossing partitions of an annulus by composing known correspondences (Barkley-Speyer for TITOs/biclosed sets, Digne-Reading for annular partitions, and Barkley for cyclic noncrossing arc diagrams) and exhibiting a suitable subset of arc diagrams.

Significance. If the pattern-avoidance characterization extends correctly and the bijection is verified, the work supplies a combinatorial model for sortable elements and noncrossing partitions in affine type A that mirrors the finite-type case. The explicit use of external bijections from Barkley-Speyer and Digne-Reading, together with the new arc-diagram subset, provides a clean chain of combinatorial equivalences without introducing fitted parameters.

major comments (2)
  1. [Definition of c-sortable biclosed sets] The section defining c-sortable biclosed sets (immediately after the pattern-avoidance characterization of affine permutations): the claim that avoidance of the listed patterns on the TITO exactly recovers the biclosed sets that are c-sortable under the affine Coxeter action is load-bearing for both the generalization and the subsequent bijection. The manuscript must supply an explicit argument (or exhaustive check on generators) showing that the finite-type patterns lift without adding or omitting affine-periodic configurations; the current reliance on the TITO bijection alone leaves this equivalence unverified.
  2. [Bijection via arc diagrams] The bijection construction via cyclic noncrossing arc diagrams: while the three external correspondences are cited, the paper must demonstrate that the subset of arc diagrams corresponding to pattern-avoiding TITOs is closed under the relevant operations and maps bijectively onto the c-noncrossing partitions without circular appeal to the original sortable definition.
minor comments (2)
  1. [Preliminaries on TITOs] Notation for TITOs and the precise list of avoided patterns should be collected in a single preliminary subsection for easier reference when reading the definition of c-sortable biclosed sets.
  2. [Abstract and introduction] The abstract states that the TITO 'coincides with the usual one-line notation in the case of an affine permutation'; a short sentence confirming that the pattern-avoidance condition reduces to the known finite-type condition on this overlap would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each major comment below and will incorporate the necessary clarifications and arguments into the revised version of the manuscript.

read point-by-point responses

  1. Referee: [Definition of c-sortable biclosed sets] The section defining c-sortable biclosed sets (immediately after the pattern-avoidance characterization of affine permutations): the claim that avoidance of the listed patterns on the TITO exactly recovers the biclosed sets that are c-sortable under the affine Coxeter action is load-bearing for both the generalization and the subsequent bijection. The manuscript must supply an explicit argument (or exhaustive check on generators) showing that the finite-type patterns lift without adding or omitting affine-periodic configurations; the current reliance on the TITO bijection alone leaves this equivalence unverified.

    Authors: We agree that an explicit verification is necessary to confirm that the pattern avoidance criterion extends correctly from the finite to the affine case. In the revised manuscript, we will add a new subsection or paragraph immediately following the pattern-avoidance characterization, providing a detailed argument. This will include showing how the finite-type patterns lift to the periodic setting of TITOs on Z, with an exhaustive check on the generators of the affine symmetric group to ensure that no extraneous affine-periodic configurations are introduced or valid ones omitted. This strengthens the foundation for defining c-sortable biclosed sets via pattern avoidance. revision: yes

  2. Referee: [Bijection via arc diagrams] The bijection construction via cyclic noncrossing arc diagrams: while the three external correspondences are cited, the paper must demonstrate that the subset of arc diagrams corresponding to pattern-avoiding TITOs is closed under the relevant operations and maps bijectively onto the c-noncrossing partitions without circular appeal to the original sortable definition.

    Authors: We will revise the bijection section to explicitly verify the required properties. Specifically, we will show that the subset of cyclic noncrossing arc diagrams arising from pattern-avoiding TITOs is closed under the operations corresponding to the affine Coxeter action and the annular partition structure. The bijection to c-noncrossing partitions will be established directly through the composition of the Barkley-Speyer correspondence (TITOs to biclosed sets), the Digne-Reading bijection (to annular partitions), and Barkley's arc diagram correspondence, using the combinatorial characterization of the subset via pattern avoidance. This avoids any circularity by grounding the argument in the properties of the arc diagrams and the external bijections rather than presupposing the sortable definition. revision: yes

Circularity Check

0 steps flagged

No circularity: generalization built from external bijections and pattern criteria without self-referential reduction

full rationale

The paper's chain begins with the external TITO representation of biclosed sets (Barkley-Speyer), applies a pattern-avoidance characterization that mirrors the finite-type case on one-line notation, defines the sortable biclosed sets via that same avoidance on TITOs, and constructs the bijection to annular noncrossing partitions through two further external correspondences (Digne-Reading and Barkley arc diagrams). None of these steps reduce a derived object to a fitted parameter, a self-citation, or a definitional tautology; each link is an independent combinatorial equivalence imported from prior literature. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces the new notion of c-sortable biclosed sets defined by pattern avoidance on TITOs; it relies on standard properties of the affine symmetric group and on three external bijections whose proofs are not reproduced here.

axioms (2)
  • domain assumption Biclosed sets of reflections in the affine symmetric group admit a translation-invariant total order (TITO) that coincides with the one-line notation of the corresponding affine permutation.
    Invoked when the paper states that TITOs coincide with the usual one-line notation, citing Barkley and Speyer.
  • standard math The affine symmetric group is a Coxeter group whose reflections and length function behave as in the finite case for the purposes of biclosed sets.
    Background assumption used throughout the generalization from finite to affine type A.
invented entities (1)
  • c-sortable biclosed set no independent evidence
    purpose: Generalization of c-sortable elements to the affine setting via pattern-avoiding TITOs.
    Newly defined object that is the central combinatorial object of the paper.

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