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Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that $\pi_{ij}=0$ is allowed. We say a plane partition is $k$-bounded if $\pi_{ij}\leq k$ for all $i,j$.

Question: What are some ways of generating uniformly random $k$-bounded weak reverse plane partitions of fixed shape $\lambda$?

I was thinking that the Hillman-Grassl correspondence would be useful here: every such partition bijects to a function $f:\lambda\rightarrow\mathbb{N}$. However, there are two things that are unclear to me here. First, is there an obvious restriction on $f$ to give $k$-bounded weak reverse partitions? Second, the bijective aspect of the correspondence gives two relations:

$$|\pi|=\sum_{v\in\lambda}f(v)h(v),$$

where $|\pi|=\sum_{ij}\pi_{ij}$ and $h(v)$ is the hook-number of the cell $v$. Equivalently, summing over all partitions,

$$\sum_{\pi}q^{|\pi|}=\prod_{v\in\lambda}\frac{1}{[h(v)]_q},$$

where $[\cdot]$ is the $q$-analogue. In both of these equations, I fail to see how one would obtain a uniform measure on the all such partitions. Is there maybe a hook-walk-like algorithm that could generate the appropriate class of $f$'s? Or perhaps there is a direct way to generate the $\pi_{ij}$?

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  • $\begingroup$ Consider the following operation: to all boxes on row $r$, add the constant $r$. This will give you a semi-standard Young tableau, if I am not mistaken, and here, you have lots of knowledge. $\endgroup$ – Per Alexandersson Feb 11 '14 at 19:01
  • $\begingroup$ @Per Alexandersson: it seems that the resulting semi standard young tableau needs to satisfy similar bounding criterion, now for each row. Unfortunately I don't see how something like a modified hook walk would give what I want. $\endgroup$ – Alex R. Feb 12 '14 at 5:15
  • $\begingroup$ Alex R: Ah, yes, you are right. But, since this corresponds to SSYT:s you probably need a modified hook-content formula. $\endgroup$ – Per Alexandersson Feb 12 '14 at 7:25

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