3 problems

Let $k\geq 1$ and let $\lambda,\mu^0,\ldots,\mu^k$ be partitions of size $n\geq 1$. A connected rooted $k$-constellation with profile $(\lambda,\mu^0,\ldots,\mu^k)$ is denoted by…

Let $k\geq 1$ and let $\lambda,\mu^0,\ldots,\mu^k$ be partitions of size $n\geq 1$. Let $\mathfrak{F}^{\lambda}_{\mu^0,\ldots,\mu^k}$ denote the set of matching tuples with the ind…

Let $k\geq 1$ and let $\lambda,\mu^0,\ldots,\mu^k$ be partitions of size $n\geq 1$. Define $c^\lambda_{\mu^0,\ldots,\mu^k}(b)$ and $h^\lambda_{\mu^0,\ldots,\mu^k}(b)$ by the coeffi…