Pattern avoidance in alternating sign rectangles I: Extended avoidance
Abstract
We introduce extendable pattern avoidance for alternating sign rectangles (ASRs), the natural rectangular generalization of alternating sign matrices (ASMs). An ASR extendably avoids a pattern π if it is the upper left corner of an ASM avoiding π. For each of the four length-three patterns in the equivalence class {312, 132, 213, 231} we establish a complete system of recurrence relations enumerating extendably π-avoiding ASRs of size r × k with a prescribed number d of nonempty rows. For π= 312 we further conjecture a closed-form expression and prove it on several diagonal slices via bijections involving Schröder ballot numbers, refined Schröder numbers and Delannoy paths that do not cross the main diagonal vertically. For ASRs of size (r−1) × (r+1) extendably avoiding $213$ we give a bijection to little Schröder paths of length r. The remaining patterns of length three, $123$ and $321$, are more elusive, mirroring the situation of ASMs.
Related
- Limiting shape of alternating sign matrices
- Return-Growth Sequences: Euler Enumeration and Pattern Avoidance
- A note on near alternating sign matrices with prescribed row and column weights
- The Colomo-Pronko conjecture for frozen-corner alternating sign matrices
- The Airy line ensemble at the edge of uniform alternating sign matrices
BibTeX
@misc{hongesberg2026pattern,
title = {{Pattern avoidance in alternating sign rectangles I: Extended avoidance}},
author = {Hans Höngesberg and Matjaž Konvalinka and Svante Linusson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07442},
eprint = {2610.07442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07442},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pattern avoidance in alternating sign rectangles I: Extended avoidance
AU - Hans Höngesberg
AU - Matjaž Konvalinka
AU - Svante Linusson
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07442
UR - https://arxiv.org/abs/2610.07442
AB - We introduce extendable pattern avoidance for alternating sign rectangles (ASRs), the natural rectangular generalization of alternating sign matrices (ASMs). An ASR extendably avoids a pattern π if it is the upper left corner of an ASM avoiding π. For each of the four length-three patterns in the equivalence class {312, 132, 213, 231} we establish a complete system of recurrence relations enumerating extendably π-avoiding ASRs of size r × k with a prescribed number d of nonempty rows. For π= 312 we further conjecture a closed-form expression and prove it on several diagonal slices via bijections involving Schröder ballot numbers, refined Schröder numbers and Delannoy paths that do not cross the main diagonal vertically. For ASRs of size (r−1) × (r+1) extendably avoiding $213$ we give a bijection to little Schröder paths of length r. The remaining patterns of length three, $123$ and $321$, are more elusive, mirroring the situation of ASMs.
ER -