{"paper":{"title":"Combinatorics of Poincar\\'e's and Schr\\\"oder's equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fr\\'ed\\'eric Menous, Jean-Christophe Novelli, Jean-Yves Thibon","submitted_at":"2015-06-26T15:05:45Z","abstract_excerpt":"We investigate the combinatorial properties of the functional equation $\\phi[h(z)]=h(qz)$ for the conjugation of a formal diffeomorphism $\\phi$ of $\\mathbb{C}$ to its linear part $z\\mapsto qz$. This is done by interpreting the functional equation in terms of symmetric functions, and then lifting it to noncommutative symmetric functions. We describe explicitly the expansion of the solution in terms of plane trees and prove that its expression on the ribbon basis has coefficients in ${\\mathbb N}[q]$ after clearing the denominators $(q)_n$. We show that the conjugacy equation can be lifted to a q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.08107","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}