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The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker\u2013Campbell\u2013Hausdorff formula due to A. Murua.<\/jats:p>","DOI":"10.1007\/s10208-021-09512-0","type":"journal-article","created":{"date-parts":[[2021,6,2]],"date-time":"2021-06-02T20:03:22Z","timestamp":1622664202000},"page":"733-755","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Cumulant\u2013Cumulant Relations in Free Probability Theory from Magnus\u2019 Expansion"],"prefix":"10.1007","volume":"22","author":[{"given":"Adrian","family":"Celestino","sequence":"first","affiliation":[]},{"given":"Kurusch","family":"Ebrahimi-Fard","sequence":"additional","affiliation":[]},{"given":"Fr\u00e9d\u00e9ric","family":"Patras","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Perales","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,2]]},"reference":[{"key":"9512_CR1","doi-asserted-by":"publisher","first-page":"1650","DOI":"10.1007\/BF01084595","volume":"17","author":"A Agrachev","year":"1981","unstructured":"Agrachev, A., Gamkrelidze, R.: Chronological algebras and nonstationary vector fields, J.\u00a0Sov.\u00a0Math.\u00a017, 1650-1675 (1981).","journal-title":"J. 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