Shin–Zeng's enumeration conjecture for signed Simsun permutations

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Let RSIn{{\mathcal{RS}}}{{\textrm{I}}}_n be the set of signed permutations whose underlying permutation is Simsun, and let RSIn(B){{\mathcal{RS}}}{{\textrm{I}}}^{(B)}_n be the subset satisfying the positivity condition on right-to-left minima in absolute value. For 1≤k≤n1\le k\le n, define

RSIn,k(B):={σ∈RSIn(B):σn=k}.{{\mathcal{RS}}}{{\textrm{I}}}^{(B)}_{n,k}:=\{\sigma\in{{\mathcal{RS}}}{{\textrm{I}}}^{(B)}_n:\sigma_n=k\}.

Shin–Zeng's conjecture. For 1≤k≤n1\le k\le n,

vn,k=#RSIn,n−k+1(B).v_{n,k}=\#{{\mathcal{RS}}}{{\textrm{I}}}^{(B)}_{n,n-k+1}.

This is an enumerative question concerning the numbers vn,kv_{n,k} and signed Simsun permutations of type I; the supplied text does not state whether the conjecture has been resolved.

References

Primary source

Sen-Peng Eu and Tung-Shan Fu, “Springer Numbers and Arnold Families Revisited”, arXiv:2111.00888 (2021).

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