Minimum support size for all achievable ranking laws

Determine the smallest function k(n) such that every achievable ranking law on the symmetric group S_n can be realized by finitely supported measures whose supports each have size at most k(n), and determine the asymptotic growth of k(n).

Background

The paper defines An\mathcal{A}_n as the set of ranking laws of nn independent, almost surely non-tied real-valued random variables. It also defines An,k\mathcal{A}_{n,k} as the subset of laws realizable by finitely supported measures with support size at most kk. Prior work cited in the paper shows that every achievable law has some finitely supported realization, but does not determine a uniform minimum support bound.

The paper constructs a full-dimensional subset of An\mathcal{A}_n using measures with support size at most ⌊e(n−1)!⌋\lfloor e(n-1)!\rfloor. The open problem asks for the optimal bound valid for all achievable laws and for its asymptotic behavior.

References

What is the smallest $k=k(n)$ such that $\AA_n = \AA_{n,k}$? What is the asymptotic growth of $k(n)$?

— Permutations from ranking independent random variables  (2610.11895 - Turvey, 8 Oct 2026) in Section 'Open Problems', item 1

What is $\dim \AA_{n,k}$ for each $k$? Remark $\ref{rem:support_bound}$ gives that $\dim \AA_{n,k} = \dim \AA_n$ for $k \geq \lfloor e(n-1)! \rfloor$. Can this upper bound on the least such $k$ be improved?

— Permutations from ranking independent random variables  (2610.11895 - Turvey, 8 Oct 2026) in Section 'Open Problems', item 2

Given any law $p \in \AA_n$, can one give an explicit algorithm that constructs finitely supported measures $\mu_1,\ldots,\mu_n$ whose ranking law is $p$? What is the computational complexity of such an algorithm? The construction in this paper achieves this only for a certain full dimensional subset of $\AA_n$.

— Permutations from ranking independent random variables  (2610.11895 - Turvey, 8 Oct 2026) in Section 'Open Problems', item 3