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permutation composition in *permutation bar* space, can it be expressed?

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For a permutation $piin S_n$ we define its bar permutation $barpi$ as:
$$barpi=[barpi(1),...barpi(n)]colon; barpi(i)=#{jle icolon pi(j)gepi(i)}$$
If we define bar space as $B_n=[1]times[2]times ...[n-1]times[n]$, the bar operation is a one-to-one map from $S_n$ to $B_n$ (see problem 3.16 in "Combinatorial Problems and Exercises" by Lovász). $B_n$ has the advantage that it decomposes the permutation to independent components (say if permutation is uniformly chosen at random its components in $B_n$ space are indpendent)

Moroever, for two permutations $pi,sigmain S_n$ permutation composition is defined as:
$$sigmacircpi = [sigma(pi(1)),sigma(pi(2)),...,sigma(pi(n))]$$

Is it possible to express composition in the bar space, i.e., to compute $overline{sigmacircpi}$ as a function of $barsigma$ and $barpi$?

combinatorics permutations

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edited 2 days ago

kvphxga

asked 2 days ago

kvphxgakvphxga

1567

$endgroup$

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  Have a look at Lehmer code en.wikipedia.org/wiki/Lehmer_code#The_code ... How is the Lehmer codes of the composition of permutations related to the Lehmer codes of the original permutations ? ... Great Question !
  $endgroup$
  – Donald Splutterwit
  2 days ago
  

  
  
  
  


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  @DonaldSplutterwit thanks! Lehmer's code is clearly tightly related to bar operation here. In fact, if we make the inequalities to strict inequality in definition of $barsigma$, and set $sigma'=[n+1-sigma(n),n+1-sigma(n-1),...,n+1-sigma(1)]$ then $barsigma=L(sigma')$
  $endgroup$
  – kvphxga
  2 days ago
  
  
  

  
  
  
  


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  I think you're going to have a tough time for general $pi$. It should however be doable for the case when $pi$ is an adjacent transposition $(i,i+1)$
  $endgroup$
  – Alex R.
  2 days ago
  

  
  
  
  


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  $begingroup$
  @AlexR. how about a combination of Lehmer's code and bar space? Like if we have $L(sigma)$ and $barpi$ or vice versa? My goal is to work with the composition in a decomposed space (that components are independent of each other). Doesn't matter what kind of decomposition though.
  $endgroup$
  – kvphxga
  2 days ago
  

  
  
  
  

add a comment | 

2

1

$begingroup$

For a permutation $piin S_n$ we define its bar permutation $barpi$ as:
$$barpi=[barpi(1),...barpi(n)]colon; barpi(i)=#{jle icolon pi(j)gepi(i)}$$
If we define bar space as $B_n=[1]times[2]times ...[n-1]times[n]$, the bar operation is a one-to-one map from $S_n$ to $B_n$ (see problem 3.16 in "Combinatorial Problems and Exercises" by Lovász). $B_n$ has the advantage that it decomposes the permutation to independent components (say if permutation is uniformly chosen at random its components in $B_n$ space are indpendent)

Moroever, for two permutations $pi,sigmain S_n$ permutation composition is defined as:
$$sigmacircpi = [sigma(pi(1)),sigma(pi(2)),...,sigma(pi(n))]$$

Is it possible to express composition in the bar space, i.e., to compute $overline{sigmacircpi}$ as a function of $barsigma$ and $barpi$?

combinatorics permutations

share|cite|improve this question

edited 2 days ago

kvphxga

asked 2 days ago

kvphxgakvphxga

1567

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Have a look at Lehmer code en.wikipedia.org/wiki/Lehmer_code#The_code ... How is the Lehmer codes of the composition of permutations related to the Lehmer codes of the original permutations ? ... Great Question !
  $endgroup$
  – Donald Splutterwit
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DonaldSplutterwit thanks! Lehmer's code is clearly tightly related to bar operation here. In fact, if we make the inequalities to strict inequality in definition of $barsigma$, and set $sigma'=[n+1-sigma(n),n+1-sigma(n-1),...,n+1-sigma(1)]$ then $barsigma=L(sigma')$
  $endgroup$
  – kvphxga
  2 days ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I think you're going to have a tough time for general $pi$. It should however be doable for the case when $pi$ is an adjacent transposition $(i,i+1)$
  $endgroup$
  – Alex R.
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @AlexR. how about a combination of Lehmer's code and bar space? Like if we have $L(sigma)$ and $barpi$ or vice versa? My goal is to work with the composition in a decomposed space (that components are independent of each other). Doesn't matter what kind of decomposition though.
  $endgroup$
  – kvphxga
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

For a permutation $piin S_n$ we define its bar permutation $barpi$ as:
$$barpi=[barpi(1),...barpi(n)]colon; barpi(i)=#{jle icolon pi(j)gepi(i)}$$
If we define bar space as $B_n=[1]times[2]times ...[n-1]times[n]$, the bar operation is a one-to-one map from $S_n$ to $B_n$ (see problem 3.16 in "Combinatorial Problems and Exercises" by Lovász). $B_n$ has the advantage that it decomposes the permutation to independent components (say if permutation is uniformly chosen at random its components in $B_n$ space are indpendent)

Moroever, for two permutations $pi,sigmain S_n$ permutation composition is defined as:
$$sigmacircpi = [sigma(pi(1)),sigma(pi(2)),...,sigma(pi(n))]$$

Is it possible to express composition in the bar space, i.e., to compute $overline{sigmacircpi}$ as a function of $barsigma$ and $barpi$?

combinatorics permutations

share|cite|improve this question

edited 2 days ago

kvphxga

asked 2 days ago

kvphxgakvphxga

1567

$endgroup$

share|cite|improve this question

edited 2 days ago

kvphxga

asked 2 days ago

kvphxgakvphxga

1567

share|cite|improve this question

edited 2 days ago

kvphxga

asked 2 days ago

kvphxgakvphxga

1567

asked 2 days ago

kvphxgakvphxga

1567

asked 2 days ago

kvphxgakvphxga

1567