MathDB

Theta conjecture

Conjecture 9.1 proposes the identity

ΘerΘek∇en−r−k∣q=1=∑P∈LD(0,n)∗k,∘rtarea(P)xP,\left.\Theta_{e_r}\Theta_{e_k}\nabla e_{n-r-k}\right\rvert_{q=1}=\sum_{P\in\mathsf{LD}(0,n)^{\ast k,\circ r}}t^{\mathsf{area}(P)}x^P,

where LD(0,n)∗k,∘r\mathsf{LD}(0,n)^{\ast k,\circ r} is the set of positively labelled Dyck paths of size nn with kk decorated rises and rr decorated contractible valleys.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

algebraic-combinatoricsformal-methodsqt-combinatorics

Progress summary

Refreshed
Open

The conjecture remains open: its special case at q=1q=1 is known, but no proof of the full two-parameter identity has been found.

D’Adderio, Iraci, and Wyngaerd formulated the Theta conjecture as Conjecture 9.1 in 2019. It predicts that a Theta-operator expression equals a generating function for labelled Dyck paths; the source explicitly leaves the full q,tq,t refinement open.

Known results

  • D’Adderio, Iraci, and Wyngaerd (2019): established the relevant q=1q=1 labelled-Dyck-path identity experimentally and proved the algebraic identity Θk∇en−k=Δen−k−1′en\Theta_k\nabla e_{n-k}=\Delta'_{e_{n-k-1}}e_n.
  • Iraci and Romero (2024): obtained related Theta-operator and decorated-path expansions, agreeing at specializations such as q=1q=1, but not the full conjecture.
  • A later refinement paper: proved implications among related touching Delta-conjecture formulations, without proving the original Theta conjecture.

Current status (as of August 2026): The q=1q=1 specialization and related algebraic identities are known, but the full labelled-Dyck-path q,tq,t identity remains open, with no reported proof, counterexample, or verification.

Sources

Solutions 0

No solutions have been posted yet.