Inserting rim-hooks into reverse plane partitions

A new algorithm for inserting rim-hooks into reverse plane partitions is presented. The insertion is used to define a bijection between reverse plane partitions of a fixed shape and multi-sets of rim-hooks. In turn this yields a bijective proof of the fact that the generating function for reverse plane partitions of a fixed shape, which was first obtained by R. Stanley, factors into a product featuring the hook-lengths of this shape. Our bijection turns out to be equivalent to a map defined by I. Pak by different means, and can be related to the Hillman–Grassl correspondence and the Robinson–Schensted–Knuth correspondence.

Keywords

bijection, reverse plane partitions, RSK

2010 Mathematics Subject Classification

05A19

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Research supported by the Austrian Science Fund (FWF), grant S50-N15 in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics” (SFB F50). The author was partially supported by the Department of Mathematics at KTH.

Received 21 May 2018

Published 14 January 2020