The Borwein conjecture and partitions with prescribed hook differences
نویسنده
چکیده
Peter Borwein has conjectured that certain polynomials have non-negative coefficients. In this paper we look at some generalizations of this conjecture and observe how they relate to the study of generating functions for partitions with prescribed hook differences. A combinatorial proof of the generating function for partitions with prescribed hook differences is given.
منابع مشابه
Some Combinatorial Properties of Hook Lengths , Contents , and Parts of Partitions 1 Richard
Citation Stanley, Richard. " Some combinatorial properties of hook lengths, contents, and parts of partitions. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract The main result of this paper is a generalization of a conjecture of Guoniu Han, originally inspired by an identity of Nekrasov and Okounkov. Our result state...
متن کاملSome Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions
The main result of this paper is a generalization of a conjecture of Guoniu Han, originally inspired by an identity of Nekrasov and Okounkov. Our result states that if F is any symmetric function (say over Q) and if
متن کاملProof of a conjecture of Okada
We prove an equation conjectured by Okada regarding hook-lengths of partitions, namely that 1 n! ∑ λ⊢n f λ ∑ u∈λ r ∏ i=1 (hu − i) = 1 2(r + 1)2 ( 2r r )( 2r + 2 r + 1 ) r ∏ j=0 (n− j), where fλ is the number of standard Young tableaux of shape λ and hu is the hook length of the square u of λ. We also obtain other similar formulas.
متن کاملGiuga's Conjecture on Primality
k=l Fermat's little theorem says that if p is a prime, then kP-l 3 lmodp for k = 1,.. .,p-1. Therefore, for each prime p, Sp--lmodp. The question becomes: Does there xist a non-prime n such that Sn 3 1 mod n? This question has resisted solution for more than forty years. After surveying what is known about the conjecture, we will give several new results here which might suggest directions of f...
متن کاملGarsia-Haiman modules for hook partitions and Green polynomials with two variables
The Garsia-Haiman modules are doubly graded modules for the symmetric groups, introduced by A. Garsia and M. Haiman [GH] to prove Macdonald’s positivity conjecture [M1]. These modules are defined for partitions of positive integers n, denoted by Dμ. The dimension of Dμ is given by n! whenever μ is a partition of n [H2]. As this fact implies, the Garsia-Haiman modules Dμ are isomorphic to the co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 3 شماره
صفحات -
تاریخ انتشار 1996