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This is a continuation of Worpitzky-like identities?. Let $ r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$

As Sam Hopkins has remarked $r_k(x)$ is the number of plane partitions in a $ \lfloor k/2 \rfloor \times \lceil k/2 \rceil \times x$ box.

By a direct computation it can be shown that $r_k(x)= \det \left( {\binom{x+ \lfloor k/2 \rfloor+i+j}{ \lfloor k/2 \rfloor +j}} \right)_{i,j = 0}^{ \lfloor (k-1)/2 \rfloor }$.

Is this a lucky coincidence or is there a combinatorial reason for this determinant?

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    $\begingroup$ Is this not just what you get from the standard way of representing plane partitions as non-intersecting lattice paths and applying Lindstrom-Gessel-Viennot? $\endgroup$
    – Sam Hopkins
    Commented Nov 15, 2021 at 16:28
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    $\begingroup$ A general comment about the series of questions you have asked: there are some facts which follow from general theory of P-partitions as developed by Stanley and which apply to all posets; then there are some facts which are "two-dimensional" in nature and have to do with nonintersecting lattice paths, determinants, etc. and which apply only to special families of posets. Combining the two categories of facts (in the ways you have done) does lead to interesting conclusions... $\endgroup$
    – Sam Hopkins
    Commented Nov 15, 2021 at 16:31

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