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PreprintCC BY 4.0arXiv

Excedances and Deficiencies of Integer Partitions

  • George E. Andrews
  • Brian Hopkins
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Abstract

The notion of a fixed point, applied by Blecher and Knopfmacher to integer partitions, can be considered as a permutation statistic. Here we apply two related permutation statistics, excedance and deficiency, to integer partitions. Using analytic and combinatorial techniques, we establish several generating functions and make connections to partition concepts including Dyson's crank statistic, overpartitions, and the Rogers--Ramanujan partitions.

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Cite

BibTeXRIS
BibTeX
@misc{andrews2026excedances,
  title = {{Excedances and Deficiencies of Integer Partitions}},
  author = {George E. Andrews and Brian Hopkins},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00435},
  eprint = {2611.00435},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00435},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Excedances and Deficiencies of Integer Partitions
AU  - George E. Andrews
AU  - Brian Hopkins
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00435
UR  - https://arxiv.org/abs/2611.00435
AB  - The notion of a fixed point, applied by Blecher and Knopfmacher to integer partitions, can be considered as a permutation statistic. Here we apply two related permutation statistics, excedance and deficiency, to integer partitions. Using analytic and combinatorial techniques, we establish several generating functions and make connections to partition concepts including Dyson's crank statistic, overpartitions, and the Rogers--Ramanujan partitions.
ER  -