A proof of the Auluck-Chowla-Gupta Conjecture

Abstract

Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n >= 1. For n >= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n <= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k -> p(n,k) fails for every n >= 4, the argument works with first differences and never uses log-concavity.

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BibTeX RIS

BibTeX

@misc{padhi2026proof,
  title = {{A proof of the Auluck-Chowla-Gupta Conjecture}},
  author = {Jaideep Sai Padhi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14084},
  eprint = {2610.14084},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14084},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - A proof of the Auluck-Chowla-Gupta Conjecture
AU  - Jaideep Sai Padhi
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.14084
UR  - https://arxiv.org/abs/2610.14084
AB  - Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n >= 1. For n >= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n <= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k -> p(n,k) fails for every n >= 4, the argument works with first differences and never uses log-concavity.
ER  -

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