Skew and direct sums of permutations

In <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, the skew sum and direct sum of <a href="/facts/Permutations/JSw9ukmv">permutations</a> are two operations to combine shorter permutations into longer ones. Given a permutation π of length m and the permutation σ of length n, the skew sum of π and σ is the permutation of length m + n defined by

(
        π
        ⊖
        σ
        )
        (
        i
        )
        =
        
          
            {
            
              
                
                  π
                  (
                  i
                  )
                  +
                  n
                
                
                  
                    for 
                  
                  1
                  ≤
                  i
                  ≤
                  m
                  ,
                
              
              
                
                  σ
                  (
                  i
                  −
                  m
                  )
                
                
                  
                    for 
                  
                  m
                  +
                  1
                  ≤
                  i
                  ≤
                  m
                  +
                  n
                  ,
                
              
            
            
          
        
      
    
    {\displaystyle (\pi \ominus \sigma )(i)={\begin{cases}\pi (i)+n&{\text{for }}1\leq i\leq m,\\\sigma (i-m)&{\text{for }}m+1\leq i\leq m+n,\end{cases}}}

and the direct sum of π and σ is the permutation of length m + n defined by

(
        π
        ⊕
        σ
        )
        (
        i
        )
        =
        
          
            {
            
              
                
                  π
                  (
                  i
                  )
                
                
                  
                    for 
                  
                  1
                  ≤
                  i
                  ≤
                  m
                  ,
                
              
              
                
                  σ
                  (
                  i
                  −
                  m
                  )
                  +
                  m
                
                
                  
                    for 
                  
                  m
                  +
                  1
                  ≤
                  i
                  ≤
                  m
                  +
                  n
                  .
                
              
            
            
          
        
      
    
    {\displaystyle (\pi \oplus \sigma )(i)={\begin{cases}\pi (i)&{\text{for }}1\leq i\leq m,\\\sigma (i-m)+m&{\text{for }}m+1\leq i\leq m+n.\end{cases}}}

Skew and direct sums of permutations open-in-new

Skew and direct sums of permutations