Connecting descent and peak polynomials
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ID: 283454
2018
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Abstract
A permutation $\sigma=\sigma_1 \sigma_2 \cdots \sigma_n$ has a descent at $i$
if $\sigma_i>\sigma_{i+1}$. A descent $i$ is called a peak if $i>1$ and $i-1$
is not a descent. The size of the set of all permutations of $n$ with a given
descent set is a polynomials in $n$, called the polynomial. Similarly, the size
of the set of all permutations of $n$ with a given peak set, adjusted by a
power of $2$ gives a polynomial in $n$, called the peak polynomial. In this
work we give a unitary expansion of descent polynomials in terms of peak
polynomials. Then we use this expansion to give a combinatorial interpretation
of the coefficients of the peak polynomial in a binomial basis, thus giving a
new proof of the peak polynomial positivity conjecture.
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oğuz2018connecting
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| Authors | Ezgi Kantarci Oğuz |
| Journal | arXiv |
| Year | 2018 |
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