Kathy O'Hara's Constructive Proof of the Unimodality of the Gaussian Polynomials
نویسنده
چکیده
The best way to learn a topic is by teaching it. Similarly, the best way to understand a new proof is by writing an expository paper about it. This was the original motivation of the present paper. Once written, I thought that it would be nice if all the readers of this Monthly had the opportunity to savor this elegant proof. All the central ideas and constructions are O’Hara’s, but I have made a few minor improvements and shortcuts that I believe to make the argument clearer. I would like to thank Kathy O’Hara for stimulating conversations and correspondence.
منابع مشابه
Unimodality of generalized Gaussian coefficients
A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients [ N λ ] q based on the explicit formula for Kostka-Foulkes polynomials is given. 1. Let us mention that the proof of the unimodality of the generalized Gaussian coefficients based on theoretic-representation considerations was given by E.B. Dynkin [1] (see also [2], [10], [11]). Recently K.O’Hara [6] gave a cons...
متن کاملUnimodality and the Reeection Principle
We show how lattice paths and the re ection principle can be used to give easy proofs of unimodality results. In particular, we give a \one-line" combinatorial proof of the unimodality of the binomial coe cients. Other examples include products of binomial coe cients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.
متن کاملUnimodality and the Reflection Principle
We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a “one-line” combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.
متن کاملInterpolation of the tabular functions with fuzzy input and fuzzy output
In this paper, rst a design is proposed for representing fuzzy polynomials withinput fuzzy and output fuzzy. Then, we sketch a constructive proof for existenceof such polynomial which can be fuzzy interpolation polynomial in a set given ofdiscrete points rather than a fuzzy function. Finally, to illustrate some numericalexamples are solved.
متن کامل