4
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.