Abstract
The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.
Highlights
1.1 ForewordThe classical hook-length formula is a beautiful result that is both mysterious and extremely well studied
Discovered by Frame, Robinson and Thrall [FRT] in 1954, it has numerous proofs, such as probabilistic, bijective, inductive, etc. In a way it is a perfect result in enumerative combinatorics – clean, concise and generalizing several others Still, the real nature of the formula remains unexplained, making potential generalizations an interesting challenge
Naruse’s formula [Naru] is a remarkable formula generalizing the hook-length formula as a sum of “hook products” over certain excited diagrams
Summary
Recall the (Semi) Standard Young tableaux and reverse plane partitions of straight and skew shape. The number f λ of standard Young tableaux of shape λ is given by the celebrated hook-length formula: Theorem 1.1 (Frame-Robinson-Thrall [FRT]) For λ a partition of n we have fλ =. We give an expression for the generating series of reverse plane partitions of skew shape in terms of a richer class of diagrams called pleasant diagrams. These diagrams are defined as the supports of arrays that are mapped to RPP of shape λ/μ by the inverse Hillman-Grassl correspondence. As a corollary of this result we derive combinatorially Naruse’s formula (1.2) (see [MPP, §6])
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