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Entry and position statistics in alternating permutations and Young tableaux

  • Ran Pan
  • Dun Qiu
  • Weichen Wang
  • Xinhui Wang
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Abstract

We study entry and position statistics in alternating permutations and standard Young tableaux using linear extensions and generating functions. For alternating permutations, we give explicit formulas for the mean and variance of the first entry and of the position of any fixed value. For standard Young tableaux, we express the sum of the lower-left entries as the number of tableaux of an associated skew shape. For two-row shapes, we determine the distribution of the entry in each cell of the second row, and for the shape (n,n) we give a closed formula for its sum over all tableaux, which recovers a limit formula of Stanley. Through a bijection of Lewis, the results for the shape (n,n) describe the entries of a random $321$-avoiding down-up permutation of even length.

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Cite

BibTeXRIS
BibTeX
@misc{pan2026entry,
  title = {{Entry and position statistics in alternating permutations and Young tableaux}},
  author = {Ran Pan and Dun Qiu and Weichen Wang and Xinhui Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00872},
  eprint = {2611.00872},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00872},
}
RIS
TY  - UNPB
TI  - Entry and position statistics in alternating permutations and Young tableaux
AU  - Ran Pan
AU  - Dun Qiu
AU  - Weichen Wang
AU  - Xinhui Wang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00872
UR  - https://arxiv.org/abs/2611.00872
AB  - We study entry and position statistics in alternating permutations and standard Young tableaux using linear extensions and generating functions. For alternating permutations, we give explicit formulas for the mean and variance of the first entry and of the position of any fixed value. For standard Young tableaux, we express the sum of the lower-left entries as the number of tableaux of an associated skew shape. For two-row shapes, we determine the distribution of the entry in each cell of the second row, and for the shape (n,n) we give a closed formula for its sum over all tableaux, which recovers a limit formula of Stanley. Through a bijection of Lewis, the results for the shape (n,n) describe the entries of a random $321$-avoiding down-up permutation of even length.
ER  -