- FindStat

Identifier

St000545: Permutations ⟶ ℤ

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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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.

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

Created

Jul 12, 2016 at 13:51 by Sara Billey

Updated

Dec 30, 2016 at 10:32 by Christian Stump