Mixing
Yet another problem of terminology about permutations
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I am looking for notations and ways to formalize applications/functions involving permutations.
Let a sequence of integers $1$, $2$, ... $n$. How to formalize, mathematically, a permutation that consists to randomly permute the $n-1$ first integers, letting unchanged the last one, $n$.
Thank you.
probability probability-theory permutations
asked Jan 13 at 20:15
Adam54Adam54
895
$endgroup$
add a comment |
$begingroup$
I am looking for notations and ways to formalize applications/functions involving permutations.
Let a sequence of integers $1$, $2$, ... $n$. How to formalize, mathematically, a permutation that consists to randomly permute the $n-1$ first integers, letting unchanged the last one, $n$.
Thank you.
probability probability-theory permutations
asked Jan 13 at 20:15
Adam54Adam54
895
$endgroup$
add a comment |
$begingroup$
I am looking for notations and ways to formalize applications/functions involving permutations.
Let a sequence of integers $1$, $2$, ... $n$. How to formalize, mathematically, a permutation that consists to randomly permute the $n-1$ first integers, letting unchanged the last one, $n$.
Thank you.
probability probability-theory permutations
asked Jan 13 at 20:15
Adam54Adam54
895
$endgroup$
I am looking for notations and ways to formalize applications/functions involving permutations.
Let a sequence of integers $1$, $2$, ... $n$. How to formalize, mathematically, a permutation that consists to randomly permute the $n-1$ first integers, letting unchanged the last one, $n$.
Thank you.
probability probability-theory permutations
probability probability-theory permutations
asked Jan 13 at 20:15
Adam54Adam54
895
asked Jan 13 at 20:15
Adam54Adam54
895
asked Jan 13 at 20:15
Adam54Adam54
895
asked Jan 13 at 20:15
Adam54Adam54
895
asked Jan 13 at 20:15
Adam54Adam54
895
895
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
This (PDF) does an excellent job of going over the various notations for permutations.
In this case, you've already formalised it. However, what you're doing is choosing a random element of $S_{n-1}$, taking the (first) natural inclusion of $S_{n-1}$ in $S_n$, and taking the image of your random element under that inclusion (equivalently, you're choosing a random element of the stabiliser $mathrm{Stab}_{S_n}(n)$ of $n$ inside $S_n$).
answered Jan 13 at 20:19
user3482749user3482749
4,266919
$endgroup$
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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$begingroup$
This (PDF) does an excellent job of going over the various notations for permutations.
In this case, you've already formalised it. However, what you're doing is choosing a random element of $S_{n-1}$, taking the (first) natural inclusion of $S_{n-1}$ in $S_n$, and taking the image of your random element under that inclusion (equivalently, you're choosing a random element of the stabiliser $mathrm{Stab}_{S_n}(n)$ of $n$ inside $S_n$).
answered Jan 13 at 20:19
user3482749user3482749
4,266919
$endgroup$
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
add a comment |
$begingroup$
This (PDF) does an excellent job of going over the various notations for permutations.
In this case, you've already formalised it. However, what you're doing is choosing a random element of $S_{n-1}$, taking the (first) natural inclusion of $S_{n-1}$ in $S_n$, and taking the image of your random element under that inclusion (equivalently, you're choosing a random element of the stabiliser $mathrm{Stab}_{S_n}(n)$ of $n$ inside $S_n$).
answered Jan 13 at 20:19
user3482749user3482749
4,266919
$endgroup$
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
add a comment |
$begingroup$
This (PDF) does an excellent job of going over the various notations for permutations.
In this case, you've already formalised it. However, what you're doing is choosing a random element of $S_{n-1}$, taking the (first) natural inclusion of $S_{n-1}$ in $S_n$, and taking the image of your random element under that inclusion (equivalently, you're choosing a random element of the stabiliser $mathrm{Stab}_{S_n}(n)$ of $n$ inside $S_n$).
answered Jan 13 at 20:19
user3482749user3482749
4,266919
$endgroup$
This (PDF) does an excellent job of going over the various notations for permutations.
In this case, you've already formalised it. However, what you're doing is choosing a random element of $S_{n-1}$, taking the (first) natural inclusion of $S_{n-1}$ in $S_n$, and taking the image of your random element under that inclusion (equivalently, you're choosing a random element of the stabiliser $mathrm{Stab}_{S_n}(n)$ of $n$ inside $S_n$).
answered Jan 13 at 20:19
user3482749user3482749
4,266919
answered Jan 13 at 20:19
user3482749user3482749
4,266919
answered Jan 13 at 20:19
user3482749user3482749
4,266919
answered Jan 13 at 20:19
user3482749user3482749
4,266919
4,266919
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
add a comment |
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
Thank you for your answer and the link. So, the searched term is "stabiliser"? I have not found the term in the linked pdf.
$endgroup$
– Adam54
Jan 13 at 20:24
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
$begingroup$
The linked PDF doesn't take the stabiliser approach: it goes for the other one instead. Specifically: there's a map from $S_{n-1}$ to $S_n$ given by taken every permutation in $S_{n-1}$ and extending it to a permutation of ${1,ldots,n}$ by just leaving $n$ unchanged. You're simply selecting an element of $S_{n-1}$, then passing it through that map. Of course, the image of that map is isomorphic to $S_{n-1}$, so for most purposes, you might as well just stick to $S_{n-1}$.
$endgroup$
– user3482749
Jan 13 at 23:10
add a comment |
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