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PreprintCC BY 4.0arXiv

Staircase diagrams, pattern avoidance, and smooth permutations

  • Faqruddin Ali Azam
  • Edward Richmond
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Abstract

A well known result due to Lakshmibai and Sandhya states that smooth Schubert varieties of type A correspond to permutations that avoid the patterns 3412 and 4231. It was later shown by the second author and Slofstra that staircase diagrams over a simply-laced Dynkin diagram are in bijection with smooth Schubert varieties of the corresponding type. We explore how the poset structure of staircase diagrams is connected with pattern avoidance on smooth permutations. As an application, we enumerate several subclasses of smooth permutations which are characterized by pattern avoidance. These subclasses include the class of polished permutations which were studied by Gaetz and Gao.

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Cite

BibTeXRIS
BibTeX
@misc{azam2026staircase,
  title = {{Staircase diagrams, pattern avoidance, and smooth permutations}},
  author = {Faqruddin Ali Azam and Edward Richmond},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33015},
  eprint = {2609.33015},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33015},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Staircase diagrams, pattern avoidance, and smooth permutations
AU  - Faqruddin Ali Azam
AU  - Edward Richmond
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.33015
UR  - https://arxiv.org/abs/2609.33015
AB  - A well known result due to Lakshmibai and Sandhya states that smooth Schubert varieties of type A correspond to permutations that avoid the patterns 3412 and 4231. It was later shown by the second author and Slofstra that staircase diagrams over a simply-laced Dynkin diagram are in bijection with smooth Schubert varieties of the corresponding type. We explore how the poset structure of staircase diagrams is connected with pattern avoidance on smooth permutations. As an application, we enumerate several subclasses of smooth permutations which are characterized by pattern avoidance. These subclasses include the class of polished permutations which were studied by Gaetz and Gao.
ER  -