Stanley symmetric function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and especially in <a href="/facts/Algebraic_combinatorics/9b47n3Fh">algebraic combinatorics</a>, the Stanley symmetric functions are a family of <a href="/facts/Symmetric_polynomials/GNhguVcn">symmetric functions</a> introduced by <a href="/facts/Richard_P._Stanley/8bQ4QrLM">Richard Stanley</a> (1984) in his study of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> of <a href="/facts/Permutation/JSw9ukmv">permutations</a>. 
Formally, the Stanley symmetric function Fw(x1, x2, ...) indexed by a permutation w is defined as a sum of certain <a href="/facts/Quasisymmetric_function/VSh6oTCV">fundamental quasisymmetric functions</a>. Each summand corresponds to a reduced decomposition of w, that is, to a way of writing w as a product of a minimal possible number of <a href="/facts/Transposition_(mathematics)/UndoYWpl">adjacent transpositions</a>. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in <a href="/facts/One-line_notation/JSw9ukmv">one-line notation</a>) has exactly

(
                  
                  
                    n
                    2
                  
                  
                    )
                  
                
              
              !
            
            
              
                1
                
                  n
                  −
                  1
                
              
              ⋅
              
                3
                
                  n
                  −
                  2
                
              
              ⋅
              
                5
                
                  n
                  −
                  3
                
              
              ⋯
              (
              2
              n
              −
              3
              
                )
                
                  1
                
              
            
          
        
      
    
    {\displaystyle {\frac {{\binom {n}{2}}!}{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots (2n-3)^{1}}}}

reduced decompositions. (Here 
 
 
 
 
 
 
 (
 
 
 n
 2
 
 
 )
 
 
 
 
 
 {\displaystyle {\binom {n}{2}}}
 
 denotes the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> n(n − 1)/2 and ! denotes the <a href="/facts/Factorial/kkfy1Bwe">factorial</a>.)

Stanley symmetric function open-in-new

Stanley symmetric function