Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,1,0,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 8
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 8
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 4
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 6
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 8
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000264
(load all 3 compositions to match this statistic)
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [[1]]
 => [1] => ([],1)
 => ? = 0 - 5
[1,2] => [[1,2]]
 => [2] => ([],2)
 => ? = 0 - 5
[2,1] => [[1],[2]]
 => [2] => ([],2)
 => ? = 2 - 5
[1,2,3] => [[1,2,3]]
 => [3] => ([],3)
 => ? = 0 - 5
[1,3,2] => [[1,2],[3]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 - 5
[2,1,3] => [[1,3],[2]]
 => [3] => ([],3)
 => ? = 4 - 5
[2,3,1] => [[1,2],[3]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2 - 5
[3,1,2] => [[1,3],[2]]
 => [3] => ([],3)
 => ? = 4 - 5
[1,2,3,4] => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = 0 - 5
[1,2,4,3] => [[1,2,3],[4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 5
[1,3,2,4] => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 5
[1,3,4,2] => [[1,2,3],[4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 5
[1,4,2,3] => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 5
[2,1,3,4] => [[1,3,4],[2]]
 => [4] => ([],4)
 => ? = 6 - 5
[2,3,1,4] => [[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 5
[2,3,4,1] => [[1,2,3],[4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 - 5
[2,4,1,3] => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 5
[3,1,2,4] => [[1,3,4],[2]]
 => [4] => ([],4)
 => ? = 6 - 5
[3,4,1,2] => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 5
[4,1,2,3] => [[1,3,4],[2]]
 => [4] => ([],4)
 => ? = 6 - 5
[1,2,3,4,5] => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = 0 - 5
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 5
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 5
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 5
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[1,5,2,3,4] => [[1,2,4,5],[3]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[2,1,3,4,5] => [[1,3,4,5],[2]]
 => [5] => ([],5)
 => ? = 8 - 5
[2,1,3,5,4] => [[1,3,4],[2,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 8 - 5
[2,1,4,5,3] => [[1,3,4],[2,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 8 - 5
[2,3,1,4,5] => [[1,2,4,5],[3]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[2,3,4,1,5] => [[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[2,3,4,5,1] => [[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 - 5
[2,3,5,1,4] => [[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[2,4,1,3,5] => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[2,4,5,1,3] => [[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[2,5,1,3,4] => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[3,1,2,4,5] => [[1,3,4,5],[2]]
 => [5] => ([],5)
 => ? = 8 - 5
[3,1,2,5,4] => [[1,3,4],[2,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 8 - 5
[3,1,4,5,2] => [[1,3,4],[2,5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 8 - 5
[3,2,4,5,1] => [[1,3,4],[2],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 8 - 5
[3,4,1,2,5] => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[3,4,5,1,2] => [[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 5
[3,5,1,2,4] => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 5
[4,1,2,3,5] => [[1,3,4,5],[2]]
 => [5] => ([],5)
 => ? = 8 - 5
[1,3,2,4,6,5] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,3,2,5,6,4] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,4,2,3,6,5] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,4,2,5,6,3] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,4,3,5,6,2] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,5,2,3,6,4] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,5,2,4,6,3] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,5,3,4,6,2] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,6,2,3,5,4] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,6,2,4,5,3] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[1,6,3,4,5,2] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,3,1,4,6,5] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,3,1,5,6,4] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,4,1,3,6,5] => [[1,2,5],[3,4,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,4,1,5,6,3] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,4,3,5,6,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,5,1,3,6,4] => [[1,2,5],[3,4,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,5,1,4,6,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,5,3,4,6,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,6,1,3,5,4] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,6,1,4,5,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[2,6,3,4,5,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,4,1,2,6,5] => [[1,2,5],[3,4,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,4,1,5,6,2] => [[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,4,2,5,6,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,5,1,2,6,4] => [[1,2,5],[3,4,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,5,1,4,6,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,5,2,4,6,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,6,1,2,5,4] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,6,1,4,5,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[3,6,2,4,5,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[4,5,1,2,6,3] => [[1,2,5],[3,4,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[4,5,1,3,6,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[4,5,2,3,6,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[4,6,1,2,5,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[4,6,1,3,5,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[4,6,2,3,5,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[5,6,1,2,4,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[5,6,1,3,4,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
[5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 8 - 5
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001629
(load all 3 compositions to match this statistic)
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 17% ●values known / values provided: 19%●distinct values known / distinct values provided: 17%
Values
[1] => [[1]]
 => [1] => [1] => ? = 0 - 7
[1,2] => [[1,2]]
 => [2] => [1] => ? = 0 - 7
[2,1] => [[1],[2]]
 => [2] => [1] => ? = 2 - 7
[1,2,3] => [[1,2,3]]
 => [3] => [1] => ? = 0 - 7
[1,3,2] => [[1,2],[3]]
 => [2,1] => [1,1] => ? = 2 - 7
[2,1,3] => [[1,3],[2]]
 => [3] => [1] => ? = 4 - 7
[2,3,1] => [[1,2],[3]]
 => [2,1] => [1,1] => ? = 2 - 7
[3,1,2] => [[1,3],[2]]
 => [3] => [1] => ? = 4 - 7
[1,2,3,4] => [[1,2,3,4]]
 => [4] => [1] => ? = 0 - 7
[1,2,4,3] => [[1,2,3],[4]]
 => [3,1] => [1,1] => ? = 2 - 7
[1,3,2,4] => [[1,2,4],[3]]
 => [2,2] => [2] => ? = 4 - 7
[1,3,4,2] => [[1,2,3],[4]]
 => [3,1] => [1,1] => ? = 2 - 7
[1,4,2,3] => [[1,2,4],[3]]
 => [2,2] => [2] => ? = 4 - 7
[2,1,3,4] => [[1,3,4],[2]]
 => [4] => [1] => ? = 6 - 7
[2,3,1,4] => [[1,2,4],[3]]
 => [2,2] => [2] => ? = 4 - 7
[2,3,4,1] => [[1,2,3],[4]]
 => [3,1] => [1,1] => ? = 2 - 7
[2,4,1,3] => [[1,2],[3,4]]
 => [2,2] => [2] => ? = 4 - 7
[3,1,2,4] => [[1,3,4],[2]]
 => [4] => [1] => ? = 6 - 7
[3,4,1,2] => [[1,2],[3,4]]
 => [2,2] => [2] => ? = 4 - 7
[4,1,2,3] => [[1,3,4],[2]]
 => [4] => [1] => ? = 6 - 7
[1,2,3,4,5] => [[1,2,3,4,5]]
 => [5] => [1] => ? = 0 - 7
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => [4,1] => [1,1] => ? = 2 - 7
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => [3,2] => [1,1] => ? = 4 - 7
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => [4,1] => [1,1] => ? = 2 - 7
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => [3,2] => [1,1] => ? = 4 - 7
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => [2,3] => [1,1] => ? = 6 - 7
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => [3,2] => [1,1] => ? = 4 - 7
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => [4,1] => [1,1] => ? = 2 - 7
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => [3,2] => [1,1] => ? = 4 - 7
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => [2,3] => [1,1] => ? = 6 - 7
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => [3,2] => [1,1] => ? = 4 - 7
[1,5,2,3,4] => [[1,2,4,5],[3]]
 => [2,3] => [1,1] => ? = 6 - 7
[2,1,3,4,5] => [[1,3,4,5],[2]]
 => [5] => [1] => ? = 8 - 7
[2,1,3,5,4] => [[1,3,4],[2,5]]
 => [4,1] => [1,1] => ? = 8 - 7
[2,1,4,5,3] => [[1,3,4],[2,5]]
 => [4,1] => [1,1] => ? = 8 - 7
[2,3,1,4,5] => [[1,2,4,5],[3]]
 => [2,3] => [1,1] => ? = 6 - 7
[2,3,4,1,5] => [[1,2,3,5],[4]]
 => [3,2] => [1,1] => ? = 4 - 7
[2,3,4,5,1] => [[1,2,3,4],[5]]
 => [4,1] => [1,1] => ? = 2 - 7
[2,3,5,1,4] => [[1,2,3],[4,5]]
 => [3,2] => [1,1] => ? = 4 - 7
[2,4,1,3,5] => [[1,2,5],[3,4]]
 => [2,3] => [1,1] => ? = 6 - 7
[2,4,5,1,3] => [[1,2,3],[4,5]]
 => [3,2] => [1,1] => ? = 4 - 7
[2,5,1,3,4] => [[1,2,5],[3,4]]
 => [2,3] => [1,1] => ? = 6 - 7
[3,1,2,4,5] => [[1,3,4,5],[2]]
 => [5] => [1] => ? = 8 - 7
[3,1,2,5,4] => [[1,3,4],[2,5]]
 => [4,1] => [1,1] => ? = 8 - 7
[3,1,4,5,2] => [[1,3,4],[2,5]]
 => [4,1] => [1,1] => ? = 8 - 7
[3,2,4,5,1] => [[1,3,4],[2],[5]]
 => [4,1] => [1,1] => ? = 8 - 7
[3,4,1,2,5] => [[1,2,5],[3,4]]
 => [2,3] => [1,1] => ? = 6 - 7
[3,4,5,1,2] => [[1,2,3],[4,5]]
 => [3,2] => [1,1] => ? = 4 - 7
[3,5,1,2,4] => [[1,2,5],[3,4]]
 => [2,3] => [1,1] => ? = 6 - 7
[4,1,2,3,5] => [[1,3,4,5],[2]]
 => [5] => [1] => ? = 8 - 7
[1,3,2,4,6,5] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,3,2,5,6,4] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,4,2,3,6,5] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,4,2,5,6,3] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,4,3,5,6,2] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,5,2,3,6,4] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,5,2,4,6,3] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,5,3,4,6,2] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,6,2,3,5,4] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,6,2,4,5,3] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[1,6,3,4,5,2] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,3,1,4,6,5] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,3,1,5,6,4] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,4,1,3,6,5] => [[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,4,1,5,6,3] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,4,3,5,6,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,5,1,3,6,4] => [[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,5,1,4,6,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,5,3,4,6,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,6,1,3,5,4] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,6,1,4,5,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[2,6,3,4,5,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,4,1,2,6,5] => [[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,4,1,5,6,2] => [[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,4,2,5,6,1] => [[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,5,1,2,6,4] => [[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,5,1,4,6,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,5,2,4,6,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,6,1,2,5,4] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,6,1,4,5,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[3,6,2,4,5,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[4,5,1,2,6,3] => [[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[4,5,1,3,6,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[4,5,2,3,6,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[4,6,1,2,5,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[4,6,1,3,5,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[4,6,2,3,5,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[5,6,1,2,4,3] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[5,6,1,3,4,2] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
[5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
 => [2,3,1] => [1,1,1] => 1 = 8 - 7
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001433
(load all 4 compositions to match this statistic)
Mapping
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001433: Signed permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => 2
[2,1,3] => [2,3,1] => [2,3,1] => 4
[2,3,1] => [2,1,3] => [2,1,3] => 2
[3,1,2] => [1,3,2] => [1,3,2] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,3,4,2] => [3,1,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 4
[2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 6
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 4
[2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 2
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 4
[3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 6
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 4
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 2
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[1,3,2,4,5] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 6
[1,3,4,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[1,3,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 4
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 6
[1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 6
[2,1,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 8
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 8
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 8
[2,3,1,4,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 6
[2,3,4,1,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 4
[2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 4
[2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 6
[2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 4
[2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 6
[3,1,2,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 8
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 8
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 8
[3,2,4,5,1] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 8
[3,4,1,2,5] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 6
[3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 4
[3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 6
[4,1,2,3,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 8
[4,1,2,5,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 8
[4,1,3,5,2] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 8
[4,2,3,5,1] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 8
[4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 6
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 8
[5,1,2,4,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 8
[5,1,3,4,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 8
[5,2,3,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 8
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,2,3,4,6,5] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2
[1,2,3,5,4,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => ? = 4
[1,2,3,5,6,4] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 2
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 6
[1,2,4,5,3,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ? = 4
[1,2,4,5,6,3] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 2
[1,2,4,6,3,5] => [4,6,1,2,3,5] => [4,6,1,2,3,5] => ? = 4
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ? = 6
[1,2,5,6,3,4] => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 4
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 6
Description
The flag major index of a signed permutation. The flag major index of a signed permutation $\sigma$ is: $$\operatorname{fmaj}(\sigma)=\operatorname{neg}(\sigma)+2\cdot \sum_{i\in \operatorname{Des}_B(\sigma)}{i} ,$$ where $\operatorname{Des}_B(\sigma)$ is the $B$-descent set of $\sigma$; see [1, Eq.(10)]. This statistic is equidistributed with the $B$-inversions ([[St001428]]) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]).
Matching statistic: St001695
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
 => [[1],[2]]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 4
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 6
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 6
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 6
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => ? = 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[4,1,2,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[4,1,3,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[4,2,3,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[4,5,1,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[5,1,2,3,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[5,1,2,4,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[5,1,3,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => ? = 0
[1,2,3,4,6,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 2
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,3,5,6,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 2
[1,2,3,6,4,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 6
[1,2,4,5,3,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,4,5,6,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 2
[1,2,4,6,3,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 6
[1,2,5,6,3,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 6
Description
The natural comajor index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Matching statistic: St001698
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
 => [[1],[2]]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [[1,3],[2,4]]
 => 2
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 4
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 6
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 6
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 6
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => ? = 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => ? = 2
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => ? = 4
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[4,1,2,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[4,1,3,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[4,2,3,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[4,5,1,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 6
[5,1,2,3,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 8
[5,1,2,4,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[5,1,3,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 8
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => ? = 0
[1,2,3,4,6,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 2
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,3,5,6,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 2
[1,2,3,6,4,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 6
[1,2,4,5,3,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,4,5,6,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => ? = 2
[1,2,4,6,3,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 6
[1,2,5,6,3,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 4
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 6
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001819
(load all 3 compositions to match this statistic)
Mapping
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001819: Signed permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[2,1,3] => [2,3,1] => [3,2,1] => [3,2,1] => 4
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[1,3,4,2] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 4
[2,1,3,4] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 6
[2,3,1,4] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[2,3,4,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 4
[3,1,2,4] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 6
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 4
[1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 2
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 4
[1,3,2,4,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 6
[1,3,4,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 4
[1,3,4,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[1,3,5,2,4] => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 4
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 6
[1,4,5,2,3] => [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 4
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 6
[2,1,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 8
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 8
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 8
[2,3,1,4,5] => [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 6
[2,3,4,1,5] => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,3,5,1,4] => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 4
[2,4,1,3,5] => [2,4,5,1,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 6
[2,4,5,1,3] => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 4
[2,5,1,3,4] => [2,3,5,1,4] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 6
[3,1,2,4,5] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 8
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 8
[3,1,4,5,2] => [3,1,4,5,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 8
[3,2,4,5,1] => [3,2,4,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 8
[3,4,1,2,5] => [3,4,5,1,2] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 6
[3,4,5,1,2] => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 4
[3,5,1,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 6
[4,1,2,3,5] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 8
[4,1,2,5,3] => [4,1,2,5,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 8
[4,1,3,5,2] => [4,1,3,5,2] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 8
[4,2,3,5,1] => [4,2,3,5,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 8
[4,5,1,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 6
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 8
[5,1,2,4,3] => [5,1,2,4,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 8
[5,1,3,4,2] => [5,1,3,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 8
[5,2,3,4,1] => [5,2,3,4,1] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 8
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,2,3,4,6,5] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2
[1,2,3,5,4,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => ? = 4
[1,2,3,5,6,4] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 2
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 6
[1,2,4,5,3,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => ? = 4
[1,2,4,5,6,3] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 2
[1,2,4,6,3,5] => [4,6,1,2,3,5] => [6,1,2,4,3,5] => [6,1,2,4,3,5] => ? = 4
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ? = 6
[1,2,5,6,3,4] => [5,6,1,2,3,4] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 4
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 6
Description
The flag Denert index of a signed permutation. The flag Denert index of a signed permutation $\sigma$ is: $$fden(\sigma) = \operatorname{neg}(\sigma) + 2 \cdot den_B(\sigma),$$ where $den_B(\sigma) = den(perm(\sigma))$ is the Denert index of the associated permutation to $\sigma$.