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The Theta Conjecture — arXiv:2609.17744

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.17744
License (as archived): CC0-1.0
Archived: 2026-10-15T16:12:59.661Z
SHA-256 of the archived text: f30c7fa830b3ac850cc104dfb2ee17653f519d50d64b55c56eed8dffb3c42c1e
Stable URL of this page: https://metraipolis.com/documento/dc_ffd9bda856df

How to cite

The Theta Conjecture — arXiv:2609.17744 — arXiv (en), https://arxiv.org/abs/2609.17744 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ffd9bda856df (Metraipolis archive dc_ffd9bda856df, sha256 f30c7fa830b3ac850cc104dfb2ee17653f519d50d64b55c56eed8dffb3c42c1e)

BibTeX

@misc{dcffd9bda856df,
  title = {The Theta Conjecture — arXiv:2609.17744},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.17744},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ffd9bda856df; sha256 f30c7fa830b3ac850cc104dfb2ee17653f519d50d64b55c56eed8dffb3c42c1e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.17744. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We define a diagonal-inversion statistic on labeled Dyck paths carrying both decorated rises and decorated contractible valleys. This gives an explicit candidate for a bivariate refinement of the univariate Theta conjecture of D'Adderio, Iraci, and Vanden Wyngaerd, recently proved by D'Adderio, Pagaria, and the authors of this work. The univariate conjecture can be recovered from our bivariate version by setting $q=1$. The new conjecture recovers the rise and valley versions of the Delta conjecture when either decoration parameter vanishes, and provides a combinatorial interpretation of the symmetric function $\Theta_{e_l} \Theta_{e_k}\nabla e_{n-k-l}$, which is also conjectured to be the Frobenius characteristic of a certain graded module of diagonal coinvariants with two sets of commuting variables and two sets of anticommuting variables. In support of the Theta conjecture and its touching refinement, we prove its Schröder case, that is, the scalar product of the symmetric function side with $e_{n-d} h_d$ matches the combinatorial side restricted to Schröder paths. The proof uses a finer combinatorial argument than the previously known cases, leveraging two Gaussian product identities.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-17 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.17744 — 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 2026-10-15T15:38:58.217Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-17?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).