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Product evaluations of Lefschetz determinants for Grassmannians and of determinants of multinomial coefficients. (English) Zbl 0736.05004
Summary: A general result which produces product evaluations of determinants of certain raising operators for $$sl(2)$$ representations is obtained. The most combinatorially interesting cases occur for self-dual raising operators of Peck posets. Applications include the following: A nice product expression is found for the determinant of the Lefschetz duality linear transformation on the cohomology of a Grassmannian. Known product expressions for the cardinalities of two sets of plane partitions are re- derived. The appearance of rising factorials for the hooks in one of these product expressions is “explained” by the appearance of rising factorials in $$sl(2)$$ determinants. A higher dimensional generalization in a certain sense of MacMahon’s famous product enumeration result for Ferrers diagrams contained in a box is stated in the context of nonintersecting lattice paths.

##### MSC:
 05A10 Factorials, binomial coefficients, combinatorial functions 05A19 Combinatorial identities, bijective combinatorics
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##### References:
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