Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 50 min ago
PreprintarXiv licencearXiv

Symmetries on vector parking functions via bounded lattice paths

  • Wenjie Fang
  • Yang Li
  • Zhicong Lin
Open at sourcePDFDOICite

Abstract

Partly motivated by enumeration of parking functions and their variants, there is a long-standing interest in lattice paths refined by several statistics, including the notable run and return. We consider bounded lattice paths, which are bounded by a given lattice path and are related to vector parking functions, for which we generalize run to composition runs, parameterized by a composition. By constructing involutions on these paths, we establish symmetries relating composition runs to some generalized return statistics. As an application, we settle an open problem of Dai, Fu, and Qiu on rational Dyck paths. The symmetries between generalized run and return are then transferred to vector parking functions, which suggest a new notion of prime decomposition. In the special case of $(a, b)$-parking functions, we compute the generating function refined by run and pri, which is new even for classical parking functions.

Related

  1. Pattern restricted parking functions with limited preferences
    Kassie Archer, Robert Laudone
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. Characterizing k-Naples parking functions and refining bounds via obstructed parking functions
    Roger Tian
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. A new generalization of parking functions: ρ-Naples parking functions
    Luca Ferrari, Francesco Verciani
    arXivPreprintCC BY-NC-SA 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials
    Igor Kortchemski
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  5. Real-rootedness and interlacing for parking functions and Chow polynomials
    Per Alexandersson
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{fang2026symmetries,
  title = {{Symmetries on vector parking functions via bounded lattice paths}},
  author = {Wenjie Fang and Yang Li and Zhicong Lin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05665},
  eprint = {2610.05665},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05665},
}
RIS
TY  - UNPB
TI  - Symmetries on vector parking functions via bounded lattice paths
AU  - Wenjie Fang
AU  - Yang Li
AU  - Zhicong Lin
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.05665
UR  - https://arxiv.org/abs/2610.05665
AB  - Partly motivated by enumeration of parking functions and their variants, there is a long-standing interest in lattice paths refined by several statistics, including the notable run and return. We consider bounded lattice paths, which are bounded by a given lattice path and are related to vector parking functions, for which we generalize run to composition runs, parameterized by a composition. By constructing involutions on these paths, we establish symmetries relating composition runs to some generalized return statistics. As an application, we settle an open problem of Dai, Fu, and Qiu on rational Dyck paths. The symmetries between generalized run and return are then transferred to vector parking functions, which suggest a new notion of prime decomposition. In the special case of $(a, b)$-parking functions, we compute the generating function refined by run and pri, which is new even for classical parking functions.
ER  -