4 problems

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Polynomiality conjecture for double cosets of parabolic linear groups
Let $n$ be a positive integer, let $\lambda,\mu\vdash n$ , and let $q$ be a prime power. Write $\operatorname{GL}_n(\mathbb F_q)$ for the general linear group and…
Conjecturesolveddouble-cosets finite-fields finite-groupsarXiv posted 2 days ago
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The double coset conjecture for finite Coxeter groups
Let $W$ be a finite Coxeter group with simple generators $S$ . For $I,J\subset S$ , let $X_{IJ}$ be the set of distinguished representatives used in the double-coset decomposition, a…
Conjectureopendouble-cosets finite-coxeter-groupsarXiv posted 2 days ago
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Maximal-size double-coset conjecture for the Sylow–Burnside process
Let $Z$ be the total number of Sylow $p$ -double cosets in $S_n$ , and let $Z_0$ be the number of double cosets of maximal size $p^{2m}$ . Maximal-size double-coset conjecture. If…
Conjectureopendouble-cosets finite-groups symmetric-groupsarXiv posted 2 days ago
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Asymptotic conjecture for parabolic double cosets in symmetric groups
Asymptotic conjecture. There exists a constant $K$ so that
Conjectureopenasymptotics double-cosets symmetric-groupsarXiv posted 2 days ago

Polynomiality conjecture for double cosets of parabolic linear groups