Liu's conjecture on gap-level excedance and Denert statistics

About 2 years old · traced to

Let g≥1g\geq 1 and ℓ≥1\ell\geq 1, and let gexcℓg\mathsf{exc}_\ell and gdenℓg\mathsf{den}_\ell denote the gg-gap ℓ\ell-level excedance number and gg-gap ℓ\ell-level Denert statistic, respectively. Set r=g+ℓ−1r=g+\ell-1. A pair of permutation statistics is rr-Euler-Mahonian when its bivariate distribution over Sn\mathfrak{S}_n agrees with that of (rdes,rmaj)(r\mathsf{des},r\mathsf{maj}). Liu's conjecture. For all g≥1g\geq 1 and ℓ≥1\ell\geq 1, the pair (gexcℓ,gdenℓ)(g\mathsf{exc}_\ell,g\mathsf{den}_\ell) is rr-Euler-Mahonian, where r=g+ℓ−1r=g+\ell-1. The paper's abstract says that its main objective is to confirm this conjecture, and the supplied proof passage says that a bijective proof is obtained; however, no explicit status evidence field is supplied for this candidate.

References

Primary source

Kaimei Huang and Sherry H. F. Yan, “Further results on r-Euler-Mahonian statistics”, arXiv:2501.12083 (2025).

Additional references

2 papers in this index state this conjecture (2024–2025). The statement above is taken from the most recent of them; the others are arXiv:2408.04185.

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.