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PreprintCC BY 4.0arXiv

Latin Eulerian Numbers

  • Madjid Mirzavaziri
  • Daniel Yaqubi
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Abstract

We introduce \emph{Latin Eulerian numbers} LEnatop k1,…,kn, a multivariate refinement of classical Eulerian numbers counting order-n Latin squares by column ascents. We establish their fundamental symmetries, univariate marginals, and an exact multiplicity divisibility property. For the total ascent statistic Σ(L)=∑i ki(L), we prove the sharp, isolated bounds n−1 ≤ Σ(L) ≤ (n−1)2, demonstrating that the adjacent values n and (n−1)2−1 are strictly unattainable. To analyze intermediate values, we show that symbol permutations typically ignored in classical enumeration directly govern Σ via an exact formula. This insight yields explicit constructions for the interior range, motivates a unimodality conjecture for the total ascent distribution.

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Cite

BibTeXRIS
BibTeX
@misc{mirzavaziri2026latin,
  title = {{Latin Eulerian Numbers}},
  author = {Madjid Mirzavaziri and Daniel Yaqubi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25100},
  eprint = {2609.25100},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25100},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Latin Eulerian Numbers
AU  - Madjid Mirzavaziri
AU  - Daniel Yaqubi
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.25100
UR  - https://arxiv.org/abs/2609.25100
AB  - We introduce \emph{Latin Eulerian numbers} LEnatop k_1,…,k_n, a multivariate refinement of classical Eulerian numbers counting order-n Latin squares by column ascents. We establish their fundamental symmetries, univariate marginals, and an exact multiplicity divisibility property. For the total ascent statistic Σ(L)=∑_i k_i(L), we prove the sharp, isolated bounds n−1 ≤ Σ(L) ≤ (n−1)^2, demonstrating that the adjacent values n and (n−1)^2−1 are strictly unattainable. To analyze intermediate values, we show that symbol permutations typically ignored in classical enumeration directly govern Σ via an exact formula. This insight yields explicit constructions for the interior range, motivates a unimodality conjecture for the total ascent distribution.
ER  -