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Statistic identifier: St000545
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Collection: Permutations
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Description: The number of parabolic double cosets with minimal element being the given permutation.
For $w \in S_n$, this is
$$\big| W_I \tau W_J\ :\ \tau \in S_n,\ I,J \subseteq S,\ w = \min\{W_I \tau W_J\}\big|$$
where $S$ is the set of simple transpositions, $W_K$ is the parabolic subgroup generated by $K \subseteq S$, and $\min\{W_I \tau W_J\}$ is the unique minimal element in weak order in the double coset $W_I \tau W_J$.
[1] contains a combinatorial description of these parabolic double cosets which can be used to compute this statistic.
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References: [1] S. Billey, M. Konvalinka, T.K. Petersen, W. Slofstra, B. Tenner, "Parabolic double cosets in Coxeter groups", to appear in Discrete Mathematics and Theoretical Computer Science, preprint 2016
[2] Number of distinct parabolic double cosets of the symmetric group S_n. [[OEIS:A260700]]
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Code:
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Statistic values:
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 6
[1,3,2] => 4
[2,1,3] => 4
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 20
[1,2,4,3] => 12
[1,3,2,4] => 16
[1,3,4,2] => 10
[1,4,2,3] => 10
[1,4,3,2] => 4
[2,1,3,4] => 12
[2,1,4,3] => 4
[2,3,1,4] => 10
[2,3,4,1] => 6
[2,4,1,3] => 8
[2,4,3,1] => 4
[3,1,2,4] => 10
[3,1,4,2] => 8
[3,2,1,4] => 4
[3,2,4,1] => 4
[3,4,1,2] => 4
[3,4,2,1] => 2
[4,1,2,3] => 6
[4,1,3,2] => 4
[4,2,1,3] => 4
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 66
[1,2,3,5,4] => 36
[1,2,4,3,5] => 48
[1,2,4,5,3] => 31
[1,2,5,3,4] => 31
[1,2,5,4,3] => 12
[1,3,2,4,5] => 48
[1,3,2,5,4] => 16
[1,3,4,2,5] => 48
[1,3,4,5,2] => 30
[1,3,5,2,4] => 32
[1,3,5,4,2] => 16
[1,4,2,3,5] => 48
[1,4,2,5,3] => 32
[1,4,3,2,5] => 16
[1,4,3,5,2] => 16
[1,4,5,2,3] => 25
[1,4,5,3,2] => 10
[1,5,2,3,4] => 30
[1,5,2,4,3] => 16
[1,5,3,2,4] => 16
[1,5,3,4,2] => 8
[1,5,4,2,3] => 10
[1,5,4,3,2] => 4
[2,1,3,4,5] => 36
[2,1,3,5,4] => 16
[2,1,4,3,5] => 16
[2,1,4,5,3] => 10
[2,1,5,3,4] => 10
[2,1,5,4,3] => 4
[2,3,1,4,5] => 31
[2,3,1,5,4] => 10
[2,3,4,1,5] => 30
[2,3,4,5,1] => 20
[2,3,5,1,4] => 24
[2,3,5,4,1] => 12
[2,4,1,3,5] => 32
[2,4,1,5,3] => 16
[2,4,3,1,5] => 16
[2,4,3,5,1] => 16
[2,4,5,1,3] => 20
[2,4,5,3,1] => 10
[2,5,1,3,4] => 24
[2,5,1,4,3] => 8
[2,5,3,1,4] => 16
[2,5,3,4,1] => 10
[2,5,4,1,3] => 8
[2,5,4,3,1] => 4
[3,1,2,4,5] => 31
[3,1,2,5,4] => 10
[3,1,4,2,5] => 32
[3,1,4,5,2] => 24
[3,1,5,2,4] => 16
[3,1,5,4,2] => 8
[3,2,1,4,5] => 12
[3,2,1,5,4] => 4
[3,2,4,1,5] => 16
[3,2,4,5,1] => 12
[3,2,5,1,4] => 8
[3,2,5,4,1] => 4
[3,4,1,2,5] => 25
[3,4,1,5,2] => 20
[3,4,2,1,5] => 10
[3,4,2,5,1] => 10
[3,4,5,1,2] => 12
[3,4,5,2,1] => 6
[3,5,1,2,4] => 20
[3,5,1,4,2] => 16
[3,5,2,1,4] => 8
[3,5,2,4,1] => 8
[3,5,4,1,2] => 8
[3,5,4,2,1] => 4
[4,1,2,3,5] => 30
[4,1,2,5,3] => 24
[4,1,3,2,5] => 16
[4,1,3,5,2] => 16
[4,1,5,2,3] => 20
[4,1,5,3,2] => 8
[4,2,1,3,5] => 16
[4,2,1,5,3] => 8
[4,2,3,1,5] => 8
[4,2,3,5,1] => 10
[4,2,5,1,3] => 16
[4,2,5,3,1] => 8
[4,3,1,2,5] => 10
[4,3,1,5,2] => 8
[4,3,2,1,5] => 4
[4,3,2,5,1] => 4
[4,3,5,1,2] => 8
[4,3,5,2,1] => 4
[4,5,1,2,3] => 12
[4,5,1,3,2] => 8
[4,5,2,1,3] => 8
[4,5,2,3,1] => 4
[4,5,3,1,2] => 4
[4,5,3,2,1] => 2
[5,1,2,3,4] => 20
[5,1,2,4,3] => 12
[5,1,3,2,4] => 16
[5,1,3,4,2] => 10
[5,1,4,2,3] => 10
[5,1,4,3,2] => 4
[5,2,1,3,4] => 12
[5,2,1,4,3] => 4
[5,2,3,1,4] => 10
[5,2,3,4,1] => 6
[5,2,4,1,3] => 8
[5,2,4,3,1] => 4
[5,3,1,2,4] => 10
[5,3,1,4,2] => 8
[5,3,2,1,4] => 4
[5,3,2,4,1] => 4
[5,3,4,1,2] => 4
[5,3,4,2,1] => 2
[5,4,1,2,3] => 6
[5,4,1,3,2] => 4
[5,4,2,1,3] => 4
[5,4,2,3,1] => 2
[5,4,3,1,2] => 2
[5,4,3,2,1] => 1
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Created: Jul 12, 2016 at 13:51 by Sara Billey
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Last Updated: Dec 30, 2016 at 10:32 by Christian Stump