On kernels of descent statistics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909214950031360 |
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| author | Clark, William L. Zhuang, Yan |
| author_facet | Clark, William L. Zhuang, Yan |
| contents | The kernel $\mathcal{K}^{\operatorname{st}}$ of a descent statistic $\operatorname{st}$, introduced by Grinberg, is a subspace of the algebra $\operatorname{QSym}$ of quasisymmetric functions defined in terms of $\operatorname{st}$-equivalent compositions, and is an ideal of $\operatorname{QSym}$ if and only if $\operatorname{st}$ is shuffle-compatible. This paper continues the study of kernels of descent statistics, with emphasis on the peak set $\operatorname{Pk}$ and the peak number $\operatorname{pk}$. The kernel $\mathcal{K}^{\operatorname{Pk}}$ in particular is precisely the kernel of the canonical projection from $\operatorname{QSym}$ to Stembridge's algebra of peak quasisymmetric functions, and is the orthogonal complement of Nyman's peak algebra. We prove necessary and sufficient conditions for obtaining spanning sets and linear bases for the kernel $\mathcal{K}^{\operatorname{st}}$ of any descent statistic $\operatorname{st}$ in terms of fundamental quasisymmetric functions, and give characterizations of $\mathcal{K}^{\operatorname{Pk}}$ and $\mathcal{K}^{\operatorname{pk}}$ in terms of the fundamental basis and the monomial basis of $\operatorname{QSym}$. Our results imply that the peak set and peak number statistics are $M$-binomial, confirming a conjecture of Grinberg. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15809 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On kernels of descent statistics Clark, William L. Zhuang, Yan Combinatorics 05E05 (Primary), 05A05, 05C50, 15A99 (Secondary) The kernel $\mathcal{K}^{\operatorname{st}}$ of a descent statistic $\operatorname{st}$, introduced by Grinberg, is a subspace of the algebra $\operatorname{QSym}$ of quasisymmetric functions defined in terms of $\operatorname{st}$-equivalent compositions, and is an ideal of $\operatorname{QSym}$ if and only if $\operatorname{st}$ is shuffle-compatible. This paper continues the study of kernels of descent statistics, with emphasis on the peak set $\operatorname{Pk}$ and the peak number $\operatorname{pk}$. The kernel $\mathcal{K}^{\operatorname{Pk}}$ in particular is precisely the kernel of the canonical projection from $\operatorname{QSym}$ to Stembridge's algebra of peak quasisymmetric functions, and is the orthogonal complement of Nyman's peak algebra. We prove necessary and sufficient conditions for obtaining spanning sets and linear bases for the kernel $\mathcal{K}^{\operatorname{st}}$ of any descent statistic $\operatorname{st}$ in terms of fundamental quasisymmetric functions, and give characterizations of $\mathcal{K}^{\operatorname{Pk}}$ and $\mathcal{K}^{\operatorname{pk}}$ in terms of the fundamental basis and the monomial basis of $\operatorname{QSym}$. Our results imply that the peak set and peak number statistics are $M$-binomial, confirming a conjecture of Grinberg. |
| title | On kernels of descent statistics |
| topic | Combinatorics 05E05 (Primary), 05A05, 05C50, 15A99 (Secondary) |
| url | https://arxiv.org/abs/2306.15809 |