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A shortest supersequence $c_n$ of all permutations on $[n]$ has length $\Theta(n^2)$: see this question on Mathoverflow. What if we force $c_n$ to be short? How many permutations can it cover?

Let's define $\#_n(s)$ to be the number of permutations on $[n]$ that are subsequences of $s$. Let $C$ denote an algorithm that reads an integer $n$ and produces a sequence of integers from $[n]$, and has a runtime bounded by $O(n\cdot{\rm polylog}(n))$. The function $f_C(n)=\#_n C(n)$ says how good algorithm $C$ is. Let $f = \sup_C f_C$, with functions ordered by their asymptotic behavior. What is the asymptotic behavior of $f$?

Has this or a similar problem been studied?

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