Commutators, matrices and an identity of Copeland

The arXiv “Commutators, matrices and an identity of Copeland” by Darij Grinberg proves and extends an identity I proposed for a matrix computation of the partition polynomials generated by iterated multiplication of a tangent vector(g(x)D)^n,where D = d/dx and g(x) is a function or formal series.

Some background and refs are given in the body and comments of the MathOverflow question “Expansions of iterated, or nested, derivatives, or vectors–conjectured matrix computation.” And another proof was added on Oct. 14, 2019.

The exponentiation and resultant partition polynomials are central to unveiling the relationships among compositional inversion via series; the differential geometry of vector fields; solutions of evolution equations, including the inviscid Burgers’ equation and the soliton solution of the KdV equation; the enumeratuve combinatorics of analytic rooted Cayley trees, Dyck paths, the associahedra, dissections of polygons, noncrossing partitions, phylogenetic trees, among other combinatorial geometric constructs; algebraic geometry and certain moduli spaces; generalized permutahedra, Hopf algebra of monoids, and optimization (see Aguiar and Ardila ref) ; free cumulants and their associated moments in free probability; enumerative combinatorics of iterated convolutions associated to the Hirzebruch criterion for the Todd class; Koszul duality and quadratic operads; and pre-Lie algebras and Butcher series (also this post).

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