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Statistic identifier: St001684

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Collection: Permutations

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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].

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

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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 


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Statistic 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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Created: Feb 20, 2021 at 14:50 by Christian Stump

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Last Updated: Mar 06, 2023 at 18:20 by Nadia Lafreniere