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Let $p(n)$ count the number of integer partitions of $n$. I am interested in the second shifted difference

$$p(n+j) - 2p(n) + p(n-j)$$

for positive integers $j$. For $j=1$, these are related to the Dyson ranks of partitions. Are there known combinatorial (or other) interpretations for other $j$?

J123
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  • Could you elaborate on the connection to Dyson's rank statistic for the $j = 1$ case? Also, that case is OEIS entry A053445; the sequences for greater $j$ are not there. – Brian Hopkins Jan 26 '22 at 19:49
  • Sure! If N(m,n) counts the number of partitions of n with rank m, then N(m,n) - N(m,n+1) can be written as a sum over these second difference functions with j=1 (from Dyson's original results).

    The sequences for larger j are also differences of integer partitions where no part is equal to j, but this felt a little unnatural, so I was hoping that they count something else. That is, let f(j,n) count the number of partitions with no part = j. Then is is clear f(j,n) = p(n)-p(n-j), so f(j,n) - f(j,n-j) is exactly the second shifted difference

     – J123 Jan 27 '22 at 23:07
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    Thanks. I think you may have answered your own question; statistics on missing parts of partitions have become more relevant recently. A related concept is the mex of a partition, the smallest missing part (using a contraction from combinatorial game theory for Minimal EXcludant) which, for instance, connects nicely to Dysons's crank statistic (e.g, an arXiv preprint). – Brian Hopkins Jan 28 '22 at 15:47

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