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PreprintCC BY-NC-ND 4.0arXiv

On Partitions with Constant-Sum Gap Sequences

  • Halime Ömrüuzun Seyrek
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Abstract

We introduce and study \emph{constant-sum gap partitions}: for a fixed nonnegative integer C, these are partitions λ=(λ1,…,λk) whose successive differences di=λi+1−λi satisfy di+dk−i=C for all valid i. We derive the generating function for this family, treating partitions of odd and even length separately, and compute it explicitly for C=0, 1,2,3,4. We then construct explicit bijections between constant-sum gap partitions with 2m+1, respectively 2m, parts and partitions into odd, respectively even, parts in which parts below the largest part have bounded multiplicity, together with an exact weight relation between a partition and its image. Finally, we give a geometric characterization of constant-sum gap partitions in terms of their Ferrers diagrams: removing a prescribed, linearly growing number of cells from each row renders the diagram symmetric about its middle row, for both the odd- and even-length cases.

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Cite

BibTeXRIS
BibTeX
@misc{seyrek2026partitions,
  title = {{On Partitions with Constant-Sum Gap Sequences}},
  author = {Halime Ömrüuzun Seyrek},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06417},
  eprint = {2611.06417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06417},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On Partitions with Constant-Sum Gap Sequences
AU  - Halime Ömrüuzun Seyrek
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06417
UR  - https://arxiv.org/abs/2611.06417
AB  - We introduce and study \emph{constant-sum gap partitions}: for a fixed nonnegative integer C, these are partitions λ=(λ_1,…,λ_k) whose successive differences d_i=λ_i+1−λ_i satisfy d_i+d_k−i=C for all valid i. We derive the generating function for this family, treating partitions of odd and even length separately, and compute it explicitly for C=0, 1,2,3,4. We then construct explicit bijections between constant-sum gap partitions with 2m+1, respectively 2m, parts and partitions into odd, respectively even, parts in which parts below the largest part have bounded multiplicity, together with an exact weight relation between a partition and its image. Finally, we give a geometric characterization of constant-sum gap partitions in terms of their Ferrers diagrams: removing a prescribed, linearly growing number of cells from each row renders the diagram symmetric about its middle row, for both the odd- and even-length cases.
ER  -