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Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...

Motivation: I'm interested in understanding the role that noncrossing partitions play in Hopf algebras/monoids (HAs) as the components of the power series of the compositional inverse of formal ...

$\newcommand{\F}{{\mathbb F}}$ $\newcommand{\Z}{{\mathbb Z}}$ Suppose that $\F$ is a finite field of prime order $p:=|\F|$, and let $n$ be a positive integer. I consider the regime where $\F$ is ...

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as $$ v_{m,...