Connecting descent and peak polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866913778585567232 |
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| author | Oğuz, Ezgi Kantarci |
| author_facet | Oğuz, Ezgi Kantarci |
| contents | A permutation $σ=σ_1 σ_2 \cdots σ_n$ has a descent at $i$ if $σ_i>σ_{i+1}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_05353 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Connecting descent and peak polynomials Oğuz, Ezgi Kantarci Combinatorics A permutation $σ=σ_1 σ_2 \cdots σ_n$ has a descent at $i$ if $σ_i>σ_{i+1}$. 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. |
| title | Connecting descent and peak polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1806.05353 |