All Questions

Let $S_n$ be the group of $n$-permutations. Denote the number of inversions of $\sigma\in S_n$ by $\ell(\sigma)$. QUESTION. Assume $n>2$. Does this cancellation property hold true? $$\sum_{\...

It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$. Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is ...

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-...

Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$. How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...