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I am essentially just re-asking this question, as it's now over a decade old, and I'm hoping that more extensive lists exist. I've started looking at the papers cited in the previous question, and have begun playing around with the Atlas software, but I'm wondering how far these efforts will get me.

For the computations I'm trying to run, Sage already is already timing out (to be fair, on a poor quality laptop) when it comes to Kazhdan–Lusztig–Vogan polynomials for the symmetric group $S_8$. I would really love to be able to make use of KLV polynomials for say $S_{16}$. Is there any hope this could be made feasible? Do lists of such polynomials exist? Naturally I'm interested in types B, C, and D as well.

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    $\begingroup$ How are you expecting to 'receive' the polynomials for $S_{16}$? Unless I'm misreading, there should be roughly $10^{26}$ of them, which is rather more than could be managed for any storage system... $\endgroup$
    – Steven Stadnicki
    Commented May 7 at 2:58
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    $\begingroup$ It seems like even people who are using machine learning/AI to study KL polynomials have only been looking at symmetric groups up to about $S_9$: see doi.org/10.1038/s41586-021-04086-x $\endgroup$
    – Sam Hopkins
    Commented May 7 at 3:07
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    $\begingroup$ @StevenStadnicki Ah, I didn't realize how quickly they explode, that dashes my hopes for big lists. Luckily, I have no interest in storing the full list, I only need/want to be able to produce polynomials P_{y, w} for specific pairs of elements y, w in S_{16} (or other S_n), and would only need to store (relatively) small lists at a time. Although if computing specific polynomial "on demand" is the only way, this also might not be feasible. $\endgroup$
    – Kristaps John Balodis
    Commented May 7 at 3:21

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