e-basis Coefficients of Chromatic Symmetric Functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910945756839936 |
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| author | Crew, Logan Zhang, Yongxing |
| author_facet | Crew, Logan Zhang, Yongxing |
| contents | A well-known result of Stanley's shows that given a graph $G$ with chromatic symmetric function expanded into the basis of elementary symmetric functions as $X_G = \sum c_λe_λ$, the sum of the coefficients $c_λ$ for $λ$ with $λ_1' = k$ (equivalently those $λ$ with exactly $k$ parts) is equal to the number of acyclic orientations of $G$ with exactly $k$ sinks.
However, more is known. The sink sequence of an acyclic orientation of $G$ is a tuple $(s_1,\dots,s_k)$ such that $s_1$ is the number of sinks of the orientation, and recursively each $s_i$ with $i > 1$ is the number of sinks remaining after deleting the sinks contributing to $s_1,\dots,s_{i-1}$. Equivalently, the sink sequence gives the number of vertices at each level of the poset induced by the acyclic orientation.
A lesser-known follow-up result of Stanley's determines certain cases in which we can find a sum of $e$-basis coefficients that gives the number of acyclic orientations of $G$ with a given partial sink sequence. Of interest in its own right, this result also admits as a corollary a simple proof of the $e$-positivity of $X_G$ when the stability number of $G$ is $2$.
In this paper, we prove a vertex-weighted generalization of this follow-up result, and conjecture a stronger version that admits a similar combinatorial interpretation for a much larger set of $e$-coefficient sums of chromatic symmetric functions. In particular, the conjectured formula would give a combinatorial interpretation for the sum of the coefficients $c_λ$ with prescribed values of $λ_1'$ and $λ_2'$ for any unweighted claw-free graph (not necessarily an incomparability graph, as in the setting of the Stanley-Stembridge conjecture). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03803 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | e-basis Coefficients of Chromatic Symmetric Functions Crew, Logan Zhang, Yongxing Combinatorics 05E05, 05C15 A well-known result of Stanley's shows that given a graph $G$ with chromatic symmetric function expanded into the basis of elementary symmetric functions as $X_G = \sum c_λe_λ$, the sum of the coefficients $c_λ$ for $λ$ with $λ_1' = k$ (equivalently those $λ$ with exactly $k$ parts) is equal to the number of acyclic orientations of $G$ with exactly $k$ sinks. However, more is known. The sink sequence of an acyclic orientation of $G$ is a tuple $(s_1,\dots,s_k)$ such that $s_1$ is the number of sinks of the orientation, and recursively each $s_i$ with $i > 1$ is the number of sinks remaining after deleting the sinks contributing to $s_1,\dots,s_{i-1}$. Equivalently, the sink sequence gives the number of vertices at each level of the poset induced by the acyclic orientation. A lesser-known follow-up result of Stanley's determines certain cases in which we can find a sum of $e$-basis coefficients that gives the number of acyclic orientations of $G$ with a given partial sink sequence. Of interest in its own right, this result also admits as a corollary a simple proof of the $e$-positivity of $X_G$ when the stability number of $G$ is $2$. In this paper, we prove a vertex-weighted generalization of this follow-up result, and conjecture a stronger version that admits a similar combinatorial interpretation for a much larger set of $e$-coefficient sums of chromatic symmetric functions. In particular, the conjectured formula would give a combinatorial interpretation for the sum of the coefficients $c_λ$ with prescribed values of $λ_1'$ and $λ_2'$ for any unweighted claw-free graph (not necessarily an incomparability graph, as in the setting of the Stanley-Stembridge conjecture). |
| title | e-basis Coefficients of Chromatic Symmetric Functions |
| topic | Combinatorics 05E05, 05C15 |
| url | https://arxiv.org/abs/2210.03803 |