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1998•Communications of the Korean Mathematical SocietyRequires access

COMBINATORIAL PROOF FOR THE GENERALIZED SCHUR IDENTITY

Jaejin Lee

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Abstract

Let λ be a partition with all distinct parts. In this paper we give a bijection between the set (X) of pairs (equation omitted) satisfying a certain condition and the set (X) of circled permutation tableaux of shape λ on the set X, where P is a tail circled shifted rim hook tableaux of shape λ and (equation omitted) is a barred permutation on X. Specializing to the partition λ with one part, this bijection gives a combinatorial proof of the Schur identity: 2(type()) = 2n! summed over all permutation with type() O . .

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What this paper is about

Let λ be a partition with all distinct parts. In this paper we give a bijection between the set (X) of pairs (equation omitted) satisfying a certain condition and the set (X) of circled permutation tableaux of shape λ on the set X, where P is a tail circled shifted rim hook tableaux of shape λ and (equation omitted) is a barred permutation on X. Specializing to the partition λ with one part, this bijection gives a combinatorial proof of the Schur identity: 2(type()) = 2n! summed over all permutation with type() O . .

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Available abstract

Let λ be a partition with all distinct parts. In this paper we give a bijection between the set (X) of pairs (equation omitted) satisfying a certain condition and the set (X) of circled permutation tableaux of shape λ on the set X, where P is a tail circled shifted rim hook tableaux of shape λ and (equation omitted) is a barred permutation on X. Specializing to the partition λ with one part, this bijection gives a combinatorial proof of the Schur identity: 2(type()) = 2n! summed over all permutation with type() O . .

Key concepts: Bijection, Mathematics, Permutation (music), Combinatorics, Partition (number theory), Combinatorial proof, Young tableau, Identity (music)

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