Hook lengths and shifted parts of partitions

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2009/04/09 Hook lengths and shifted parts of partitions Guo-Niu HAN Dedicated to George Andrews, on the occasion of his seventieth birthday. ABSTRACT. — Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric func- tions can be used instead. The purpose of this note is to prove it. 1. Introduction The hook lengths of partitions are widely studied in the Theory of Partitions, in Algebraic Combinatorics and Group Representation Theory. The basic notions needed here can be found in [St99, p.287; La01, p.1]. A partition ? is a sequence of positive integers ? = (?1, ?2, · · · , ?) such that ?1 ≥ ?2 ≥ · · · ≥ ? > 0. The integers (?i)i=1,2,..., are called the parts of ?, the number of parts being the length of ? denoted by (?). The sum of its parts ?1+?2 + · · ·+? is denoted by |?|. Let n be an integer, a partition ? is said to be a partition of n if |?| = n.

parts ?1

?j ?

metric functions

hook lengths

finally h?

partition ?

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Andrews

2009/04/09

Hook lengths and shifted parts of partitions

GuoNiu HAN

Dedicated to George Andrews, on the occasion of his seventieth birthday.

ABSTRACTSome conjectures on partition hook lengths, recently. — stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric func tions can be used instead. The purpose of this note is to prove it.

1. Introduction The hook lengths of partitions are widely studied in the Theory of Partitions, in Algebraic Combinatorics and Group Representation Theory. The basic notions needed here can be found in [St99, p.287; La01, p.1]. A partitionλis a sequence of positive integersλ= (λ1, λ2,∙ ∙ ∙, λℓ) such that λ1≥λ2∙ ∙ ≥≥ ∙ λℓ>integers (0. The λi)i=1,2,...,ℓare called theparts ofλ, the numberℓof parts being thelengthofλdenoted byℓ(λ). The sum of its partsλ1+λ2+∙ ∙ ∙+λℓis denoted by|λ|. Letnbe an integer, a partitionλis said to be a partition ofnif|λ|=nwrite. We λ⊢n. Each partition can be represented by its Ferrers diagram. For each boxvin the Ferrers diagram of a partitionλ, or for each boxvinλ, for short, deﬁne thehook lengthofv, denoted byhv(λ) orhv, to be the number of boxesu such thatu=v, orulies in the same column asvand abovev, or in the same row asvand to the right ofvproduct of all hook lengths of. The λ is denoted byHλ.

The hook length plays an important role in Algebraic Combinatorics thanks to the famous hook formula due to Frame, Robinson and Thrall [FRT54]

(1.1)

n! fλ=, Hλ

wherefλis the number of standard Young tableaux of shapeλ.

Key words and phrases.partitions, hook lengths, hook formulas, sym metric functions. Mathematics Subject Classiﬁcations.05A15, 05A17, 05A19, 05E05, 11P81

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