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— We introduce a hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple(More)
Dédiéà Dominique Foatà a l'occasion de sonsoixantì eme anniversaire R ´ ESUM´ E. — Soit U un ensemble de couples de lettres. Foata et Zeilberger [FZ] ont introduit les U-statistiques pour les mots quelconques. Dans cette note, onétablit une condition nécessaire et suffisante pour que les deux définitions " maj U " et " maj2 U " , qu'on rencontre dans le cas(More)
Abstract. We extend Stanley’s work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used:(More)
Efficient Legendre moment computation for grey level images G.Y. Yang, H.Z. Shu, G.N. Han, C. Toumoulin, L.M. Luo Laboratory of Image Science and Technology, Department of Biology and Medical Engineering , Southeast University, 210096, Nanjing, People’s Republic of China IRMA, Université Louis Pasteur et C.N.R.S., 7, rue René-Descartes F, 67084 Strasbourg,(More)
— The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t-cores, and give an elementary proof by using the Macdonald identities. We also obtain(More)
— We find two new hook length formulas for binary trees. The particularity of our formulas is that the hook length h v appears as an exponent. Consider the set B (n) of all binary trees with n vertices. It is well-known that the cardinality of B (n) is equal to the Catalan number (see, e.g., [9, p.220]): (1) T ∈B(n) 1 = 1 n + 1 2n n. For each vertex v of a(More)
Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that a(n) = 0 if and only if b(n) = 0, where integers a(n) and b(n) are defined by
Abstract. We present some conjectures and open problems on partition hook lengths, which are all motivated by known results on the subject. The conjectures are suggested by extensive experimental calculations using a computer algebra system. The first conjecture unifies two classical results on the number of standard Young tableaux and the number of pairs(More)
We construct two bijections of the symmetric group Sn onto itself that enable us to show that three new three-variable statistics are equidistributed with classical statistics involving the number of fixed points. The first one is equidistributed with the triplet (fix, des,maj), the last two with (fix, exc,maj), where “fix,” “des,” “exc” and “maj” denote(More)