query: Every element of $S_n$ can be written as a product of transpositions of the form $(1 i )$, for various $i$
positive_passages.docid: 67926
positive_passages.text: $S_n$ is generated by elements of the form $(1k)$
negative_passages.docid: 132541
negative_passages.text: Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$
negative_passages.docid: 64527
negative_passages.text: Show that an operation of elements in a finite group contains each element exactly once
negative_passages.docid: 109440
negative_passages.text: How does $(12\dots n)$ and $(a\, b)$ generate $S_n$?
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 120642
negative_passages.text: Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
negative_passages.docid: 109710
negative_passages.text: Bijection between $\mathbb N \times \mathbb N$ and $\mathbb N$
negative_passages.docid: 1627
negative_passages.text: Cartesian product of an arbitrary number of sets
negative_passages.docid: 128966
negative_passages.text: Explicit bijection between $\Bbb R$ and permutations of $\Bbb N$
negative_passages.docid: 46971
negative_passages.text: Number of surjections from $\{1,...,m\}$ to $\{1,...,n\}$
negative_passages.docid: 96680
negative_passages.text: Generators of symmetric group $S_n$
negative_passages.docid: 118471
negative_passages.text: Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
negative_passages.docid: 74924
negative_passages.text: identity element of $(D_n ,\gcd)$
negative_passages.docid: 16476
negative_passages.text: Order of the product of two commuting elements with coprime orders in a group.
negative_passages.docid: 155372
negative_passages.text: Find the number of permutations in $S_n$ containing fixed elements in one cycle
negative_passages.docid: 15202
negative_passages.text: the root of $N$th transpose matrix
negative_passages.docid: 1507
negative_passages.text: The product of $n$ consecutive integers is divisible by $n$ factorial