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Permutations with only reduced co-BPDs

Joshua Arroyo, Adam Gregory

TL;DR

This work characterizes precisely which permutations have only reduced co-BPDs in the bumpless pipe dream framework relating Grothendieck and Schubert polynomials. It introduces the pattern set $\{1423,12543,13254,25143,215643,216543,241653\}$ and proves both directions: avoidance of these patterns by $w$ guarantees all co-BPDs are reduced, while presence of any pattern yields a non-reduced co-BPD, via configuration lemmas and explicit droop-move constructions. The approach connects geometric configurations of elbows and blocking pipes to classical permutation patterns, enabling a checkable criterion for co-BPD reduction. Consequently, for pattern-avoiding $w$, every co-BPD contributes positively to the structure of the Grothendieck–Schubert expansions, with corollaries regarding reversals and vexillary cases that illuminate the interplay between combinatorics and K-theoretic/schubert theory.

Abstract

Bumpless pipe dreams (BPDs) are combinatorial objects used in the study of Schubert and Grothendieck polynomials. Weigandt recently introduced a co-BPD object associated to each BPD and used them to give an analogue to the change of bases formulas of Lenart and Lascoux between these polynomials. She posed the problem of characterizing the set of permutations whose BPDs have only reduced co-BPDs. We give a pattern-avoidance characterization for these permutations using a set of seven patterns.

Permutations with only reduced co-BPDs

TL;DR

This work characterizes precisely which permutations have only reduced co-BPDs in the bumpless pipe dream framework relating Grothendieck and Schubert polynomials. It introduces the pattern set and proves both directions: avoidance of these patterns by guarantees all co-BPDs are reduced, while presence of any pattern yields a non-reduced co-BPD, via configuration lemmas and explicit droop-move constructions. The approach connects geometric configurations of elbows and blocking pipes to classical permutation patterns, enabling a checkable criterion for co-BPD reduction. Consequently, for pattern-avoiding , every co-BPD contributes positively to the structure of the Grothendieck–Schubert expansions, with corollaries regarding reversals and vexillary cases that illuminate the interplay between combinatorics and K-theoretic/schubert theory.

Abstract

Bumpless pipe dreams (BPDs) are combinatorial objects used in the study of Schubert and Grothendieck polynomials. Weigandt recently introduced a co-BPD object associated to each BPD and used them to give an analogue to the change of bases formulas of Lenart and Lascoux between these polynomials. She posed the problem of characterizing the set of permutations whose BPDs have only reduced co-BPDs. We give a pattern-avoidance characterization for these permutations using a set of seven patterns.
Paper Structure (7 sections, 24 theorems, 19 equations, 13 figures)

This paper contains 7 sections, 24 theorems, 19 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Configuration lemmas
  4. non-reduced implies pattern containment
  5. pattern implies non-reduced
  6. Concluding Remarks
  7. Droop move sequences

Key Result

Theorem 1.1

A permutation $w$ has all reduced co-BPDs if and only if $w$ avoids the seven patterns in $\Pi = \{ 1423, 12543, 13254, 25143, 215643, 216543, 241653 \}$.

Figures (13)

  • Figure 1: A (reduced) BPD for 1423 with its corresponding (non-reduced) co-BPD which traces out 3412.
  • Figure 2: Note that an instance of the configuration does not necessarily guarantee that the same pipes will crossing at both pipes. However, the crossing pipes can only change if there is another instance of the configuration contained within the configuration.
  • Figure 3: The above example illustrates the importance of requiring chains of alternating elbows. This BPD does not have the configuration as there is not a chain of alternating elbows and thus the co-BPD is reduced.
  • Figure 4: An example of such a pipe $r$ from Lemma \ref{['lem:oneFlat']}.
  • Figure 5: An example of such a pipe $r$ from Lemma \ref{['lem:twoFlats']}
  • ...and 8 more figures

Theorems & Definitions (54)

  • Theorem 1.1
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: lam2021back
  • Definition 2.6: weigandt2021bumpless
  • Definition 2.7: weigandt2025changingbasespipedream
  • Theorem 2.8: weigandt2025changingbasespipedream
  • ...and 44 more