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A Quasisymmetric Analogue of the Bruhat Order

  • Ryan Bansal
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Abstract

We define a quasisymmetric analogue of the Bruhat order on indexed binary forests using the multiplicity-free Monk rule for forest polynomials. For the finite window 𝒫n of forests supported in [1,n], we construct an explicit partition into saturated chains terminating at the top rank, proving that 𝒫n is strongly Sperner. We also recover the known Catalan ballot formula for its rank numbers. For principal lower intervals, we prove that [βˆ…,F] is a lattice if and only if it is Boolean, and characterize this property by the avoidance of a bent corner in F, a path of three internal vertices that changes direction. This gives a forest analogue of the pattern-avoidance characterization of Boolean principal Bruhat intervals. We also conjecture that every principal lower interval is shellable, providing computational evidence and proving the conjecture for a family of hook intervals.

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Cite

BibTeXRIS
BibTeX
@misc{bansal2026quasisymmetric,
  title = {{A Quasisymmetric Analogue of the Bruhat Order}},
  author = {Ryan Bansal},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02210},
  eprint = {2611.02210},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02210},
}
RIS
TY  - UNPB
TI  - A Quasisymmetric Analogue of the Bruhat Order
AU  - Ryan Bansal
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02210
UR  - https://arxiv.org/abs/2611.02210
AB  - We define a quasisymmetric analogue of the Bruhat order on indexed binary forests using the multiplicity-free Monk rule for forest polynomials. For the finite window 𝒫_n of forests supported in [1,n], we construct an explicit partition into saturated chains terminating at the top rank, proving that 𝒫_n is strongly Sperner. We also recover the known Catalan ballot formula for its rank numbers. For principal lower intervals, we prove that [βˆ…,F] is a lattice if and only if it is Boolean, and characterize this property by the avoidance of a bent corner in F, a path of three internal vertices that changes direction. This gives a forest analogue of the pattern-avoidance characterization of Boolean principal Bruhat intervals. We also conjecture that every principal lower interval is shellable, providing computational evidence and proving the conjecture for a family of hook intervals.
ER  -