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PreprintCC BY 4.0arXiv

Distinct flattened partitions avoiding a pattern of length four

  • Toufik Mansour
  • Olivia Nabawanda
  • Mark Shattuck
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Abstract

Let 𝒫n denote the set of distinct permutations of length n that arise from the flattening process applied to the partitions of [n]={1,…,n}. In this paper, we consider the problem of avoidance of a single classical pattern of length four by members of 𝒫n. Let pn(Ο„) denote the number of members of 𝒫n that avoid the pattern Ο„. We show that pn(Ο„)=Cnβˆ’1 for all n β‰₯ 1 for seven patterns of length four yielding new combinatorial interpretations of the Catalan number sequence. Further, we show that pn(Ο„) corresponds to the binomial transform of Catalan numbers for three other patterns. To establish our results, we suitably refine the counting sequence pn(Ο„) in each case so as to obtain a system of functional equations satisfied by the corresponding generating functions. These functional equations may then be solved explicitly leading to a determination of pn(Ο„) in each case.

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Cite

BibTeXRIS
BibTeX
@misc{mansour2026distinct,
  title = {{Distinct flattened partitions avoiding a pattern of length four}},
  author = {Toufik Mansour and Olivia Nabawanda and Mark Shattuck},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11271},
  eprint = {2610.11271},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11271},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Distinct flattened partitions avoiding a pattern of length four
AU  - Toufik Mansour
AU  - Olivia Nabawanda
AU  - Mark Shattuck
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11271
UR  - https://arxiv.org/abs/2610.11271
AB  - Let 𝒫_n denote the set of distinct permutations of length n that arise from the flattening process applied to the partitions of [n]={1,…,n}. In this paper, we consider the problem of avoidance of a single classical pattern of length four by members of 𝒫_n. Let p_n(Ο„) denote the number of members of 𝒫_n that avoid the pattern Ο„. We show that p_n(Ο„)=C_nβˆ’1 for all n β‰₯ 1 for seven patterns of length four yielding new combinatorial interpretations of the Catalan number sequence. Further, we show that p_n(Ο„) corresponds to the binomial transform of Catalan numbers for three other patterns. To establish our results, we suitably refine the counting sequence p_n(Ο„) in each case so as to obtain a system of functional equations satisfied by the corresponding generating functions. These functional equations may then be solved explicitly leading to a determination of p_n(Ο„) in each case.
ER  -