Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Hook lengths in a skew Young diagram. | Hook lengths in a skew Young diagram Svante Janson Department of Mathematics Uppsala University PO Box 480 S-751 06 Uppsala Sweden Submitted October 15 1997 Accepted October 20 1997 Abstract Regev and Vershik Electronic J. Combinatorics 4 1997 R22 have obtained some properties of the set of hook lengths for certain skew Young diagrams using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently Regev and Zeilberger Annals of Combinatorics to appear have independently proved this conjecture. 1 Introduction Regev and Vershik 1 have recently obtained some properties of the set of hook lengths for certain skew Young diagrams. They prove the results using asymptotic calculations of the degrees of certain sequences of characters of the symmetric group and note that they do not know a direct finite proof of their results. The purpose of the present note is to present such a proof for one of their results viz. their Theorem . Moreover Regev and Vershik s Theorem states that two different sets of hook lengths have the same product and the authors conjecture that in fact these two sets are equal. More precisely the sets in question should be regarded as multisets . the elements may have multiplicities. We prove this conjecture. Very recently Regev and Zeilberger 2 have independently proved this conjecture. THE ELECTRONIC JOURNAL OF COMBINATORICS 4 1997 R24 2 2 Notation If n1 n2 nm 0 are integers with m 1 let D D n1 . nm be the Young diagram with rows of lengths n1 . nm. Following and slightly extending Regev and Vershik 1 we introduce the following definitions see the examples in Figures 1 and 2. R R m n is an m X n rectangle . a Young diagram with m rows of equal length n. We assume that n n1 and that R and D are positioned such that their top left corners coincide. Then D c R. Regev and Vershik consider only the
