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Descent Tops and Pinnacles on 2143- and 3421-Avoiding Permutations
Abstract
We prove the 2143-3421 case of Burstein's conjecture on the distribution of descent-top sets by constructing an explicit bijection between the two avoidance classes. The bijection also preserves the pinnacle set and reverses the left-to-right order of the pinnacle values. We combine an involution on maximum Cartesian trees with exchanges of the final entries of maximal decreasing runs. On an intermediate avoidance class, we prove that the exchange procedures terminate and that the output in each direction is independent of the choices made. This gives an explicit inverse. Restricting the bijection to Dumont permutations of the first kind proves the conjecture of Burstein and Jones that 2143 and 3421 are Wilf-equivalent on this class
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BibTeX
@misc{dong2026descent,
title = {{Descent Tops and Pinnacles on 2143- and 3421-Avoiding Permutations}},
author = {Yue Dong and Lily Li Liu and Tongyuan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01077},
eprint = {2610.01077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01077},
}
RIS
TY - UNPB
TI - Descent Tops and Pinnacles on 2143- and 3421-Avoiding Permutations
AU - Yue Dong
AU - Lily Li Liu
AU - Tongyuan Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01077
UR - https://arxiv.org/abs/2610.01077
AB - We prove the 2143-3421 case of Burstein's conjecture on the distribution of descent-top sets by constructing an explicit bijection between the two avoidance classes. The bijection also preserves the pinnacle set and reverses the left-to-right order of the pinnacle values. We combine an involution on maximum Cartesian trees with exchanges of the final entries of maximal decreasing runs. On an intermediate avoidance class, we prove that the exchange procedures terminate and that the output in each direction is independent of the choices made. This gives an explicit inverse. Restricting the bijection to Dumont permutations of the first kind proves the conjecture of Burstein and Jones that 2143 and 3421 are Wilf-equivalent on this class
ER -