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Analysis of the asymmetric shelf shuffle

  • Raghavendra Tripathi
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Abstract

In an asymmetric shelf shuffle, a deck of n cards is dealt sequentially from the bottom and assigned one of the m shelves uniformly at random. The card is placed at the top of the assigned shelf with probability p, and at the bottom of the assigned shelf with probability (1−p). Analysis of the shelf shuffle has gained much attention recently, and the case p=1/2 was first treated by Diaconis--Fulman--Holmes [Ann. Appl. Prob. 23 (2013), no. 4, 1692--1720]. In this paper, we extend the analysis of the shelf shuffle to general p∈ (0, 1). In particular, we study the distribution of cycles, cycle lengths, number of descents, number of valleys, number of inversions, and the RSK shape of a permutation obtained from an asymmetric shelf shuffle. Our results extend the analysis of Diaconis--Fulman--Holmes to arbitrary p. Furthermore, our analysis of the distribution of descents and inversions is new even for p=1/2.

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Cite

BibTeXRIS
BibTeX
@misc{tripathi2026analysis,
  title = {{Analysis of the asymmetric shelf shuffle}},
  author = {Raghavendra Tripathi},
  howpublished = {arXiv},
  year = {2026},
  month = jun,
  doi = {10.48550/arxiv.2606.18047},
  eprint = {2606.18047},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.18047},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Analysis of the asymmetric shelf shuffle
AU  - Raghavendra Tripathi
PY  - 2026
DA  - 2026/06/16
PB  - arXiv
DO  - 10.48550/arxiv.2606.18047
UR  - https://arxiv.org/abs/2606.18047
AB  - In an asymmetric shelf shuffle, a deck of n cards is dealt sequentially from the bottom and assigned one of the m shelves uniformly at random. The card is placed at the top of the assigned shelf with probability p, and at the bottom of the assigned shelf with probability (1−p). Analysis of the shelf shuffle has gained much attention recently, and the case p=1/2 was first treated by Diaconis--Fulman--Holmes [Ann. Appl. Prob. 23 (2013), no. 4, 1692--1720]. In this paper, we extend the analysis of the shelf shuffle to general p∈ (0, 1). In particular, we study the distribution of cycles, cycle lengths, number of descents, number of valleys, number of inversions, and the RSK shape of a permutation obtained from an asymmetric shelf shuffle. Our results extend the analysis of Diaconis--Fulman--Holmes to arbitrary p. Furthermore, our analysis of the distribution of descents and inversions is new even for p=1/2.
ER  -