- FindStat

Identifier

St001684: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1]=>0
[1,2]=>0
[2,1]=>1
[1,2,3]=>0
[1,3,2]=>1
[2,1,3]=>1
[2,3,1]=>2
[3,1,2]=>2
[3,2,1]=>4
[1,2,3,4]=>0
[1,2,4,3]=>1
[1,3,2,4]=>1
[1,3,4,2]=>2
[1,4,2,3]=>2
[1,4,3,2]=>4
[2,1,3,4]=>1
[2,1,4,3]=>3
[2,3,1,4]=>2
[2,3,4,1]=>3
[2,4,1,3]=>4
[2,4,3,1]=>6
[3,1,2,4]=>2
[3,1,4,2]=>4
[3,2,1,4]=>4
[3,2,4,1]=>6
[3,4,1,2]=>5
[3,4,2,1]=>8
[4,1,2,3]=>3
[4,1,3,2]=>6
[4,2,1,3]=>6
[4,2,3,1]=>9
[4,3,1,2]=>8
[4,3,2,1]=>12
[1,2,3,4,5]=>0
[1,2,3,5,4]=>1
[1,2,4,3,5]=>1
[1,2,4,5,3]=>2
[1,2,5,3,4]=>2
[1,2,5,4,3]=>4
[1,3,2,4,5]=>1
[1,3,2,5,4]=>3
[1,3,4,2,5]=>2
[1,3,4,5,2]=>3
[1,3,5,2,4]=>4
[1,3,5,4,2]=>6
[1,4,2,3,5]=>2
[1,4,2,5,3]=>4
[1,4,3,2,5]=>4
[1,4,3,5,2]=>6
[1,4,5,2,3]=>5
[1,4,5,3,2]=>8
[1,5,2,3,4]=>3
[1,5,2,4,3]=>6
[1,5,3,2,4]=>6
[1,5,3,4,2]=>9
[1,5,4,2,3]=>8
[1,5,4,3,2]=>12
[2,1,3,4,5]=>1
[2,1,3,5,4]=>3
[2,1,4,3,5]=>3
[2,1,4,5,3]=>5
[2,1,5,3,4]=>5
[2,1,5,4,3]=>8
[2,3,1,4,5]=>2
[2,3,1,5,4]=>5
[2,3,4,1,5]=>3
[2,3,4,5,1]=>4
[2,3,5,1,4]=>6
[2,3,5,4,1]=>8
[2,4,1,3,5]=>4
[2,4,1,5,3]=>7
[2,4,3,1,5]=>6
[2,4,3,5,1]=>8
[2,4,5,1,3]=>8
[2,4,5,3,1]=>11
[2,5,1,3,4]=>6
[2,5,1,4,3]=>10
[2,5,3,1,4]=>9
[2,5,3,4,1]=>12
[2,5,4,1,3]=>12
[2,5,4,3,1]=>15
[3,1,2,4,5]=>2
[3,1,2,5,4]=>5
[3,1,4,2,5]=>4
[3,1,4,5,2]=>6
[3,1,5,2,4]=>7
[3,1,5,4,2]=>10
[3,2,1,4,5]=>4
[3,2,1,5,4]=>8
[3,2,4,1,5]=>6
[3,2,4,5,1]=>8
[3,2,5,1,4]=>10
[3,2,5,4,1]=>12
[3,4,1,2,5]=>5
[3,4,1,5,2]=>8
[3,4,2,1,5]=>8
[3,4,2,5,1]=>11
[3,4,5,1,2]=>9
[3,4,5,2,1]=>13
[3,5,1,2,4]=>8
[3,5,1,4,2]=>12
[3,5,2,1,4]=>12
[3,5,2,4,1]=>15
[3,5,4,1,2]=>14
[3,5,4,2,1]=>18
[4,1,2,3,5]=>3
[4,1,2,5,3]=>6
[4,1,3,2,5]=>6
[4,1,3,5,2]=>9
[4,1,5,2,3]=>8
[4,1,5,3,2]=>12
[4,2,1,3,5]=>6
[4,2,1,5,3]=>10
[4,2,3,1,5]=>9
[4,2,3,5,1]=>12
[4,2,5,1,3]=>12
[4,2,5,3,1]=>15
[4,3,1,2,5]=>8
[4,3,1,5,2]=>12
[4,3,2,1,5]=>12
[4,3,2,5,1]=>15
[4,3,5,1,2]=>14
[4,3,5,2,1]=>18
[4,5,1,2,3]=>9
[4,5,1,3,2]=>14
[4,5,2,1,3]=>14
[5,1,2,3,4]=>4
[5,1,2,4,3]=>8
[5,1,3,2,4]=>8
[5,1,3,4,2]=>12
[5,1,4,2,3]=>11
[5,1,4,3,2]=>15
[5,2,1,3,4]=>8
[5,2,1,4,3]=>12
[5,2,3,1,4]=>12
[5,2,3,4,1]=>15
[5,2,4,1,3]=>15
[5,3,1,2,4]=>11
[5,3,1,4,2]=>15
[5,3,2,1,4]=>15
[5,4,1,2,3]=>13
                    

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Description

The reduced word complexity of a permutation.
For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$.
For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$.
This statistic appears in [1, Question 6.1].

References

[1] Knutson, A., Miller, E. Subword complexes in Coxeter groups MathSciNet:2047852

Code

def statistic(pi): 
    ws = pi.reduced_words() 
    Ws = Words(sorted(ws[0])) 
    ws = [ Ws(w) for w in ws ] 
    n = pi.length() 
    while True: 
        for w in Ws.iterate_by_length(n): 
            if all(v.is_subword_of(w) for v in ws): 
                return n 
        n += 1

Created

Feb 20, 2021 at 14:50 by Christian Stump

Updated

Mar 06, 2023 at 18:20 by Nadia Lafreniere