Boolean, Free, and Classical Cumulants as Tree Enumerations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909308718940160 |
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| author | Defant, Colin Lee, Mitchell |
| author_facet | Defant, Colin Lee, Mitchell |
| contents | Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boolean, Free, and Classical Cumulants as Tree Enumerations Defant, Colin Lee, Mitchell Combinatorics 05A05, 05A18, 05A19, 46L53 Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes. |
| title | Boolean, Free, and Classical Cumulants as Tree Enumerations |
| topic | Combinatorics 05A05, 05A18, 05A19, 46L53 |
| url | https://arxiv.org/abs/2409.05219 |