Elementary symmetric partitions
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916453410668544 |
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| author | Ballantine, Cristina Beck, George Merca, Mircea Sagan, Bruce |
| author_facet | Ballantine, Cristina Beck, George Merca, Mircea Sagan, Bruce |
| contents | Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that pre_2 is injective as a map on binary partitions of n. In the present work we derive a host of identities involving the sequences which count the number of parts of a given value in the image of pre_2. These include generating functions, explicit expressions, and formulas for forward differences. We generalize some of these to d-ary partitions and explore connections with color partitions. Our techniques include the use of generating functions and bijections on rooted partitions. We end with a list of conjectures and a direction for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elementary symmetric partitions Ballantine, Cristina Beck, George Merca, Mircea Sagan, Bruce Combinatorics 05A17, 05E05 (Primary) 05A15, 05A19 (Secondary) Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands in the evaluation e_k(mu_1,...,mu_l). The study of such partitions was initiated by Ballantine, Beck, and Merca who showed (among other things) that pre_2 is injective as a map on binary partitions of n. In the present work we derive a host of identities involving the sequences which count the number of parts of a given value in the image of pre_2. These include generating functions, explicit expressions, and formulas for forward differences. We generalize some of these to d-ary partitions and explore connections with color partitions. Our techniques include the use of generating functions and bijections on rooted partitions. We end with a list of conjectures and a direction for future research. |
| title | Elementary symmetric partitions |
| topic | Combinatorics 05A17, 05E05 (Primary) 05A15, 05A19 (Secondary) |
| url | https://arxiv.org/abs/2409.11268 |