Halving Partitions and Hyperbinary Expansions

Abstract

Starting from a positive integer, repeatedly halve the current value, rounding up or down. Which resulting sequences have terms that add to the starting integer? A simple balance identity turns the rounding choices into hyperbinary expansions, in which each power of two occurs at most twice. This correspondence gives exact counts by length and an injection into the expansions counted by Stern's diatomic sequence. We characterize equality in the resulting bound by the binary digits of the odd part of the integer. We also determine the unique longest partition and give a short recursive counting algorithm. When the required total is allowed to vary, the chain counts eventually equal Stern's number, and we determine the exact threshold. A direct proof shows that the counts of hyperbinary expansions in the three length classes modulo three differ by at most one. For sufficiently long halving partitions, the counts by length first weakly increase and then weakly decrease, although the full sequence need not do so.

Halving the original Kalton--Roberts upper bound for nearly additive set functions
Boon Suan Ho, Tomasz Kania
arXiv28 Aug 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
The Maximum Number of Halving Lines for Finite Planar Sets—New Lower Bounds
Javier Rodrigo, Sagrario Lantarón, María Dolores López
Axioms15 Oct 2026Peer-reviewedCC BY 4.0General mathematicsCrossref
The bivariate generating function for cylindric partitions of rank three
Shane Chern, Trieu Nguyen, Chanh Tran +1
arXiv29 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
On the growth of weak regular partitions in k-uniform hypergraphs
H. Sheats, C. Terry
arXiv1 Nov 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematics Logic & foundationsarXiv
The quasisymmetric flag variety: a toric complex on noncrossing partitions
Nantel Bergeron, Lucas Gagnon, Philippe Nadeau +2
Forum of Mathematics, Pi10 Jul 2026Peer-reviewedCC BY 4.0Full textAlgebra & number theory Combinatorics & discrete mathematicsHAL

Cite

BibTeX RIS

BibTeX

@misc{alekseyev2026halving,
  title = {{Halving Partitions and Hyperbinary Expansions}},
  author = {Max A. Alekseyev},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01243},
  eprint = {2611.01243},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01243},
}

RIS

TY  - UNPB
TI  - Halving Partitions and Hyperbinary Expansions
AU  - Max A. Alekseyev
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01243
UR  - https://arxiv.org/abs/2611.01243
AB  - Starting from a positive integer, repeatedly halve the current value, rounding up or down. Which resulting sequences have terms that add to the starting integer? A simple balance identity turns the rounding choices into hyperbinary expansions, in which each power of two occurs at most twice. This correspondence gives exact counts by length and an injection into the expansions counted by Stern's diatomic sequence. We characterize equality in the resulting bound by the binary digits of the odd part of the integer. We also determine the unique longest partition and give a short recursive counting algorithm. When the required total is allowed to vary, the chain counts eventually equal Stern's number, and we determine the exact threshold. A direct proof shows that the counts of hyperbinary expansions in the three length classes modulo three differ by at most one. For sufficiently long halving partitions, the counts by length first weakly increase and then weakly decrease, although the full sequence need not do so.
ER  -

Abstract

Related

Cite