Geodesic
On the modes of polynomials derived from nondecreasing sequences
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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Wang and Yeh proved that if $P(x)$ is a polynomial with nonnegative and nondecreasing coefficients, then $P(x+d)$ is unimodal for any $d>0$. A mode of a unimodal polynomial $f(x)=a_0+a_1x+\cdots + a_mx^m$ is an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_*(P,d)$ is the smallest mode of $P(x+d)$, and $M^*(P,d)$ the greatest mode. Wang and Yeh conjectured that if $d_2>d_1>0$, then $M_*(P,d_1)\geq M_*(P,d_2)$ and $M^*(P,d_1)\geq M^*(P,d_2)$. We give a proof of this conjecture.
DOI : 10.37236/488
Classification : 05A20, 33F10
Mots-clés : unimodal polynomials, smallest mode, greatest mode 
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     author = {Donna Q. J. Dou and Arthur L. B. Yang},
     title = {On the modes of polynomials derived from nondecreasing sequences},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/488},
     zbl = {1209.05021},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/488/}
}
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Donna Q. J. Dou; Arthur L. B. Yang. On the modes of polynomials derived from nondecreasing sequences. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/488

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