Schur-positivity for generalized nets
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929643060199424 |
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| author | Shelburne, Ethan van Willigenburg, Stephanie |
| author_facet | Shelburne, Ethan van Willigenburg, Stephanie |
| contents | A graph is Schur-positive if its chromatic symmetric function expands nonnegatively in the Schur basis. All claw-free graphs are conjectured to be Schur-positive. We introduce a combinatorial object corresponding to a graph G, called a special rim hook G-tabloid, which is a variation on the special rim hook tabloid. These objects can be employed to compute any Schur coefficient of the chromatic symmetric function of a graph. We construct sign-reversing maps on these special rim hook G-tabloids to obtain a recurrence relation for the Schur coefficients of a family of claw-free graphs called generalized nets, then we prove the entire family is Schur-positive. We subsequently determine an analogous recurrence relation for another, similar family of claw-free graphs. Thus, we demonstrate a new method for proving Schur-positivity of chromatic symmetric functions, which has the potential to be applied to make further progress toward the aforementioned conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schur-positivity for generalized nets Shelburne, Ethan van Willigenburg, Stephanie Combinatorics A graph is Schur-positive if its chromatic symmetric function expands nonnegatively in the Schur basis. All claw-free graphs are conjectured to be Schur-positive. We introduce a combinatorial object corresponding to a graph G, called a special rim hook G-tabloid, which is a variation on the special rim hook tabloid. These objects can be employed to compute any Schur coefficient of the chromatic symmetric function of a graph. We construct sign-reversing maps on these special rim hook G-tabloids to obtain a recurrence relation for the Schur coefficients of a family of claw-free graphs called generalized nets, then we prove the entire family is Schur-positive. We subsequently determine an analogous recurrence relation for another, similar family of claw-free graphs. Thus, we demonstrate a new method for proving Schur-positivity of chromatic symmetric functions, which has the potential to be applied to make further progress toward the aforementioned conjecture. |
| title | Schur-positivity for generalized nets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.00943 |