Many plane partition enumeration formulae have proofs using representation theory. For example, let $PP((p)^q,m)$ denote the $p \times q$ plane partitions of height at most $m$. For $p > q$, let $PP(\nabla_{p,q},m)$ denote the plane partitions of shifted shape $\nabla_{p,q} = (p+q-1, p+q-3, \dots, p-q+1)$ of height at most $m$.Tje $(p)^q$ plane partition describe weights of irreducible representations in $sl(p+q,\mathbb{C})$ while the $\nabla_{p,q}$ plane partitions describe weights of irreducible $sp(2k,\mathbb{C})$ representations when $p +q = 2k$.In this paper, Proctor showed that$$|PP(\nabla_{p,q},k)| = |PP((p)^q,k)| = \prod^p_{i=1} \prod^q_{j=1}\prod^m_{k=1} \frac{(i+j+k-1)}{(i+j+k-2)},$$where the second equality is a well known result of MacMahon's.His proof used a branching rule to show the irreducible representations in question are equinumerous, and at present no bijective proof has been published.
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