Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials
Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials
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DOI：
10.37236/6806
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发表时间：
2016-01
期刊：
影响因子：
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通讯作者：
Herman Z. Q. Chen;A. Yang;Philip B. Zhang
中科院分区：
文献类型：
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作者：
Herman Z. Q. Chen;A. Yang;Philip B. Zhang
在Kostka数和Catalan数的研究中，Kirillov提出了矩形Narayana多项式的单峰猜想。证明了矩形Narayana多项式只有真实的零点，从而利用牛顿不等式证实了Kirillov的单峰猜想。通过使用一个equidistribution属性之间的下降数和上升数的选票路径由于Sulanke和格字和标准的杨tableaux之间的双射，我们表明，矩形Narayana多项式是等于下降生成函数的标准杨tableaux的某些矩形形状，直到幂的不定。基于Brenti关于标准Young tableaux的下降母函数只有真实的零点的结果，得到了矩形Narayana多项式的实根性.
In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture with the help of Newton's inequality. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of certain rectangular shape, up to a power of the indeterminate. Then we obtain the real-rootedness of the rectangular Narayana polynomial based on Brenti's result that the descent generating function of standard Young tableaux has only real zeros.