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Combinatorial aspects of the Delannoy Lattice

  • Xi Chen
  • Yuxian Dong
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Abstract

The Delannoy numbers d(n,k) count lattice paths from (0,0) to (nโˆ’k,k) using steps $(1,0),(0,1)$ and (1,1). This paper introduces a graded poset ๐’Ÿn on the Delannoy paths ending on the line x+y=n, whose rank-generating function is the Delannoy polynomial dn(x)=โˆ‘k=0n d(n,k)xk. We prove that mathcalDn is a self-dual lattice, which is call the Delannoy lattice. By establishing an explicit symmetric Boolean decomposition, we show that ๐’Ÿn is a symmetric Boolean order, thereby recovering the ฮณ-positivity of dn(x). Such a decomposition is refined to a symmetric chain decomposition with the chain cover property, and is applied to determine all maximum antichains. We also investigate other combinatorial aspects of ๐’Ÿn, including supersolvability, the Mรถbius number, characteristic polynomials, and zeta polynomials.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026combinatorial,
  title = {{Combinatorial aspects of the Delannoy Lattice}},
  author = {Xi Chen and Yuxian Dong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30350},
  eprint = {2609.30350},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30350},
}
RIS
TY  - UNPB
TI  - Combinatorial aspects of the Delannoy Lattice
AU  - Xi Chen
AU  - Yuxian Dong
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30350
UR  - https://arxiv.org/abs/2609.30350
AB  - The Delannoy numbers d(n,k) count lattice paths from (0,0) to (nโˆ’k,k) using steps $(1,0),(0,1)$ and (1,1). This paper introduces a graded poset ๐’Ÿ_n on the Delannoy paths ending on the line x+y=n, whose rank-generating function is the Delannoy polynomial d_n(x)=โˆ‘_k=0^n d(n,k)x^k. We prove that mathcalD_n is a self-dual lattice, which is call the Delannoy lattice. By establishing an explicit symmetric Boolean decomposition, we show that ๐’Ÿ_n is a symmetric Boolean order, thereby recovering the ฮณ-positivity of d_n(x). Such a decomposition is refined to a symmetric chain decomposition with the chain cover property, and is applied to determine all maximum antichains. We also investigate other combinatorial aspects of ๐’Ÿ_n, including supersolvability, the Mรถbius number, characteristic polynomials, and zeta polynomials.
ER  -