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Symmetry distribution between hook length and part length for partitions

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2009/04/09 Symmetry distribution between hook length and part length for partitions Christine Bessenrodt and Guo-Niu Han ABSTRACT. — It is known that the two statistics on integer partitions “hook length” and “part length” are equidistributed over the set of all partitions of n. We extend this result by proving that the bivariate joint generating function by those two statistics is symmetric. Our method is based on a generating function by a triple statistic much easier to calculate. 1. Introduction The basic notions needed here can be found in [11, p.287]. 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. We write ? n. Each partition can be represented by its Ferrers diagram (or Young diagram). For each box v in the Ferrers diagram of a partition ?, or for each box v in ?, for short, define the arm length (resp.

  • statistics hook

  • joint generating function

  • part length

  • hook lengths

  • v?? xhvypv

  • generating function

  • bivariate joint

  • all part

  • super-symmetric


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2009/04/09
Symmetry distribution between hook length and part length for partitions
Christine Bessenrodt and GuoNiu Han
ABSTRACTIt is known that the two statistics on integer partitions. — “hook length” and “part length” are equidistributed over the set of all partitions ofnextend this result by proving that the bivariate joint. We generating function by those two statistics is symmetric.Our methodis based on a generating function by a triple statistic much easier to calculate. 1. Introduction The basic notions needed here can be found in [11, p.287].Apartitionλ is a sequence of positive integersλ= (λ1, λ2,∙ ∙ ∙, λ) such thatλ1λ2∙ ∙ ∙λ>0. Theintegersλi,i= 1,2, . . . , ℓare called thepartsofλ, the numberof parts being thelengthofλdenoted by(λ). Thesum of its partsλ1+λ2+∙ ∙ ∙+λis denoted by|λ|. Letnbe an integer; a partition λis said to be a partition ofnif|λ|=nwrite. Weλn. Each partition can be represented by its Ferrers diagram (or Young diagram). Foreach boxvin the Ferrers diagram of a partitionλ, or for each boxvinλ, for short, define thearm length(resp.leg length,coarm length,coleg length) ofv, denoted byavorav(λ) (resp.lv,mv,gv), to be the number of boxesusuch thatulies in the same row asvand to the right ofvthe same column as(resp. invand abovev, in the same row as vand to the left ofv, in the same column asvand undervFig.1.). See We define thehook length(resp.part length) ofvinλto behv= av+lv(resp.+ 1pv=mv+av+ 1).Bessenrodt [2], Bacher and Manivel [3] have proved that the two statisticshvandpvare equidistributed over the set of all partitions ofn, i.e., X XX X hvpv (1)x=x . λn vλ λn vλ For example, the set of all partitions of 4 with their hook lengths (resp. part lengths) is reproduced in Fig. 2 (resp.Fig. 3).We see that the two 2 3 above generating functions byhvand bypvare identical 7x+ 6x+ 3x+ 4 4x. Key words and phrases.partitions, hook lengths, hook type, symmetry distribution. Mathematics Subject Classifications.05A15, 05A17, 05A19, 11P81 1