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During a calculation, I have met a function on partitions, $F(\lambda)$, which seems to evaluate to positive integers. I have a procedure for computing it, but I was hoping for a formula, so I thought I would put the numbers here in case someone has seen them before or has a good guess.

I had no luck with the OEIS.

For the smallest partitions, which is all I can compute really, the function is:

\begin{array}{|l|l|} \hline \lambda & F(\lambda) \\ \hline (1) & 1 \\ \hline (2) & 1 \\ \hline (1,1) & 1 \\ \hline (3) & 4 \\ \hline (2,1) & 3 \\ \hline (1,1,1) & 1 \\ \hline (4) & 20 \\ \hline (3,1) & 16 \\ \hline (2,2) & 5 \\ \hline (2,1,1) & 6 \\ \hline (1,1,1,1) & 1 \\ \hline (5) & 148 \\ \hline (4,1) & 100 \\ \hline (3,2) & 60 \\ \hline (3,1,1) & 40 \\ \hline (2,2,1) & 25 \\ \hline (2,1,1,1) & 10 \\ \hline (1,1,1,1,1) & 1 \\ \hline \end{array}

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    $\begingroup$ Does $(6)$ give $1348$? $\endgroup$
    – Peter Taylor
    Commented Dec 20, 2021 at 12:39
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    $\begingroup$ Have you tried findstat.org? $\endgroup$
    – Sam Hopkins
    Commented Dec 20, 2021 at 12:45
  • $\begingroup$ But anyways I just checked with the data you provided and as far as I can see it did not find any matches... $\endgroup$
    – Sam Hopkins
    Commented Dec 20, 2021 at 13:56
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    $\begingroup$ Maybe if you post the procedure, it can be optimised. I was speculating that the function applied to partitions into one part might correspond to OEIS A001171. $\endgroup$
    – Peter Taylor
    Commented Dec 20, 2021 at 14:25
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    $\begingroup$ excuse stating the obvious: the sum seems to be oeis.org/A000165. So it might be a map from signed permutations to integer partitions in disguise. $\endgroup$
    – Martin Rubey
    Commented Dec 20, 2021 at 17:27

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