Type B area conjecture for q,t-Fuss-Catalan numbers

About 18 years old · traced to

For n≥1n\geq 1, let WBnW_{B_n} be the reflection group of type BnB_n. A type BB Catalan path of length nn is a lattice path of 2n2n north and east steps from (0,0)(0,0) staying above the diagonal, and let Cat⁡Bn(q)\operatorname{Cat}_{B_n}(q) be the sum of qarea⁡(λ)q^{\operatorname{area}(\lambda)} over such paths. Type B area conjecture.

Cat⁡(1)(WBn,q,1)=Cat⁡Bn(q).\operatorname{Cat}^{(1)}(W_{B_n},q,1)=\operatorname{Cat}_{B_n}(q).

The conjecture defines the proposed area interpretation of the q,tq,t-Fuss-Catalan polynomial in type BB and is supported by the explicit calculations and path enumeration given in the paper. Its resolution is not specified in the supplied material.

References

Primary source

Christian Stump, “q,t-Fuss-Catalan numbers for complex reflection groups”, arXiv:0806.2936 (2008).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.