Metraipolis › archive › dc_66211656edcb

Iterated Distinct Absolute Differences of Integer Compositions — arXiv:2608.04468

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.04468
License (as archived): CC0-1.0
Archived: 2026-10-31T11:46:00.257Z
SHA-256 of the archived text: 4ca90f16c65b347496531d08a5adc4b4495789f5f4b94dc1e41f4dda98f3146b
Stable URL of this page: https://metraipolis.com/documento/dc_66211656edcb

How to cite

Iterated Distinct Absolute Differences of Integer Compositions — arXiv:2608.04468 — arXiv (en), https://arxiv.org/abs/2608.04468 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_66211656edcb (Metraipolis archive dc_66211656edcb, sha256 4ca90f16c65b347496531d08a5adc4b4495789f5f4b94dc1e41f4dda98f3146b)

BibTeX

@misc{dc66211656edcb,
  title = {Iterated Distinct Absolute Differences of Integer Compositions — arXiv:2608.04468},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.04468},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_66211656edcb; sha256 4ca90f16c65b347496531d08a5adc4b4495789f5f4b94dc1e41f4dda98f3146b},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.04468. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Starting from an integer composition, form its consecutive absolute differences, provided that they are nonzero and pairwise distinct, and then permute these differences arbitrarily before repeating the operation. The depth of the composition is the maximum possible number of successive iterations. We determine the least positive integer admitting a composition of any prescribed depth. If a(n) is the least positive integer having a composition of depth n, then $a(n)=n+1+\lceil n(n+1)/4\rceil+\lfloor n/2\rfloor$. We also prove that every integer k at least a(n) has a composition of depth at least n. Consequently, if d(k) denotes the maximum depth of a composition of k, then $d(k)=\max\{n\geq 0:a(n)\leq k\}$. Finally, we classify and enumerate all compositions attaining the minimum a(n). Their number is given by four factorial formulas according to n modulo 4.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-06 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.04468 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-31T11:35:10.101Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-06?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).