Correctly defining an application in a set of permutations

Question

Let $S_n$ be the set of permutations of $[n]$.

How to define correctly a function $f:\ S_n \rightarrow S_{n-1}$ which deletes $n$? Is it implicitly clear?

Thank you

Answer 1

From your comment, you could define $f$ as follows.

Let $\sigma$ be a permutation of $[n]=\{1,\dots,n\}$. If $\sigma$ fixes $n$, i.e. $\sigma(n)=n$. Then we define $f(\sigma)=\sigma|_{[n-1]}$ as the restriction of $\sigma$ to $\{1,\dots,n-1\}$. Otherwise $\sigma(k)=n$ for some $k<n$. We define the permutation $f(\sigma)=\sigma'$ by removing the value $n$ and attributing $\sigma(k+1)$ instead. Then we shift all values above. More precisely, we define $\sigma'(m)=\sigma(m)$ if $m<k$ and $\sigma'(m)=\sigma(m+1)$ if $m\geq k$.

With this definition, the permutation $$ \sigma=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 5 & 6 & 4 & 3 & 1 \end{pmatrix} $$ is mapped to $$ f(\sigma)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1 \end{pmatrix}. $$