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User:Tom Copeland

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My mini-arxiv with notes related mainly to entries in the OEIS, MathOverflow, and Math Stackexchange: Shadows of Simplicity at

Some of my favorite entries lying at the crossroads of geometry, combinatorics, and analysis are

1) on the multiplicative reciprocal/inverse of an exponential generating function/formal Taylor series and its connection to the geometric combinatorics of the classic permutohedra/permutahedra polytopes

2) on the refined Eulerian numbers/partition polynomials, related to compositional inversion, the h-vectors of permutahedra, and differential evolution equations/flow fields.

3) on the compositional inverse of ordinary generating functions/formal power series and its connection to the geometric combinatorics of the associahedra, or Stasheff polytopes

4) on the compositional inverse of an ordinary generating function/formal power series in terms of the power series coefficients of its scaled, multiplicative inverse and its connection to the enumeration of noncrossing partitions, free probability, and Appell Sheffer polynomial sequences

5) and, on the Faber polynomials and the cycle index polynomials of the symmetric groups (a.k.a. the Stirling partition polynomials of the first kind), related to symmetric polynomials/functions and therefore Appell sequences; traces, determinants, and other invariants of matrices under conjugation by pairs of inverse matrices; Chern characteristic classes; paths on graphs; and more.