Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000804
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000804: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => 0
[2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,1,3] => 0
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 0
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 0
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 0
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 0
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0
[2,3,1,4] => [2,4,1,3] => [1,4,2,3] => 1
[2,3,4,1] => [2,4,3,1] => [1,3,2,4] => 2
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [4,1,3,2] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => 0
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => 0
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => 0
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => 0
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,1,4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => 0
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0
Description
The number of occurrences of the vincular pattern |123 in a permutation. This is the number of occurrences of the pattern $(1,2,3)$, such that the letter matched by $1$ is the first entry of the permutation.
Matching statistic: St000802
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000802: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,3,2] => [2,3,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [2,3,1] => [1,3,2] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,2,3,4] => 0
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 0
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 1
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [4,2,3,1] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 1
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [4,2,3,1] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [1,4,2,3] => 0
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [1,4,2,3] => 0
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [3,4,1,2] => 0
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [3,4,2,1] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [1,4,3,2] => 0
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [2,4,3,1] => 0
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [1,2,4,3] => 0
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [1,2,4,3] => 0
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [3,2,4,1] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,3,4,2] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,4,5] => 0
Description
The number of occurrences of the vincular pattern |321 in a permutation. This is the number of occurrences of the pattern $(3,2,1)$, such that the letter matched by $3$ is the first entry of the permutation.
Matching statistic: St001084
Mapping
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001084: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,3,2] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 0
[3,2,1] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [4,1,3,2] => 0
[1,2,4,3] => [2,4,3,1] => [4,1,3,2] => [4,1,2,3] => 0
[1,3,2,4] => [3,2,4,1] => [4,2,1,3] => [4,2,3,1] => 0
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => [4,3,2,1] => 0
[1,4,2,3] => [3,4,2,1] => [4,3,1,2] => [4,2,1,3] => 0
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => [4,3,1,2] => 0
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [3,4,1,2] => 0
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [3,4,2,1] => 0
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2
[2,4,1,3] => [1,4,2,3] => [1,3,4,2] => [2,3,1,4] => 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [3,1,2,4] => 0
[3,1,4,2] => [4,1,3,2] => [2,4,3,1] => [3,2,1,4] => 0
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [2,4,3,1] => 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[3,4,1,2] => [4,1,2,3] => [2,3,4,1] => [2,1,4,3] => 0
[3,4,2,1] => [3,1,2,4] => [2,3,1,4] => [1,4,2,3] => 0
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
[4,1,3,2] => [4,3,1,2] => [3,4,2,1] => [3,2,4,1] => 0
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [1,3,4,2] => 1
[4,3,1,2] => [4,2,1,3] => [3,2,4,1] => [2,1,3,4] => 0
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => [1,4,3,2] => 0
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,4,2,3] => 0
[1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => [5,1,3,2,4] => 0
[1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => [5,1,4,3,2] => 0
[1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => [5,1,3,4,2] => 0
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,4,2,3] => [5,1,2,3,4] => 0
[1,2,5,4,3] => [2,5,4,3,1] => [5,1,4,3,2] => [5,1,2,4,3] => 0
[1,3,2,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => [5,2,4,1,3] => 0
[1,3,2,5,4] => [3,2,5,4,1] => [5,2,1,4,3] => [5,2,3,1,4] => 0
[1,3,4,2,5] => [4,2,3,5,1] => [5,2,3,1,4] => [5,3,4,1,2] => 0
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [5,3,2,4,1] => 0
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [5,4,2,3,1] => 0
[1,4,2,3,5] => [3,4,2,5,1] => [5,3,1,2,4] => [5,2,4,3,1] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [5,2,3,4,1] => 0
[1,4,3,2,5] => [4,3,2,5,1] => [5,3,2,1,4] => [5,3,4,2,1] => 0
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [5,4,3,1,2] => 0
[1,4,5,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => [5,3,2,1,4] => 0
[1,4,5,3,2] => [5,4,2,3,1] => [5,3,4,2,1] => [5,4,2,1,3] => 0
Description
The number of occurrences of the vincular pattern |1-23 in a permutation. This is the number of occurrences of the pattern $123$, where the first two matched entries are the first two entries of the permutation. In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
[1,2] => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => [1,2] => ([],2)
 => ? = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 1
[2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ? = 1 + 1
[3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => ? = 0 + 1
[3,2,1] => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => ? = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 0 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 0 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 0 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => ? = 0 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => ? = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,5,3,1,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,5,6,3] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,4,6,3] => [.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St001491
(load all 8 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Values
[1,2] => [1,2] => 1 => 1 = 0 + 1
[2,1] => [2,1] => 0 => ? = 0 + 1
[1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[1,3,2] => [1,3,2] => 10 => 1 = 0 + 1
[2,1,3] => [2,1,3] => 01 => 1 = 0 + 1
[2,3,1] => [2,3,1] => 00 => ? = 1 + 1
[3,1,2] => [3,1,2] => 00 => ? = 0 + 1
[3,2,1] => [3,2,1] => 00 => ? = 0 + 1
[1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => 100 => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => 100 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => 100 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => 100 => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => 010 => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => 010 => 1 = 0 + 1
[2,3,1,4] => [2,4,1,3] => 000 => ? = 1 + 1
[2,3,4,1] => [2,4,3,1] => 000 => ? = 2 + 1
[2,4,1,3] => [2,4,1,3] => 000 => ? = 1 + 1
[2,4,3,1] => [2,4,3,1] => 000 => ? = 2 + 1
[3,1,2,4] => [3,1,4,2] => 000 => ? = 0 + 1
[3,1,4,2] => [3,1,4,2] => 000 => ? = 0 + 1
[3,2,1,4] => [3,2,1,4] => 001 => 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => 000 => ? = 1 + 1
[3,4,1,2] => [3,4,1,2] => 000 => ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => 000 => ? = 0 + 1
[4,1,2,3] => [4,1,3,2] => 000 => ? = 0 + 1
[4,1,3,2] => [4,1,3,2] => 000 => ? = 0 + 1
[4,2,1,3] => [4,2,1,3] => 000 => ? = 0 + 1
[4,2,3,1] => [4,2,3,1] => 000 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => 000 => ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => 000 => ? = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,5,2,4,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,1,3,5,4] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,1,4,3,5] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,1,4,5,3] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,1,5,3,4] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,1,5,4,3] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,3,1,4,5] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[2,3,1,5,4] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[2,3,4,1,5] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[2,3,4,5,1] => [2,5,4,3,1] => 0000 => ? = 3 + 1
[2,3,5,1,4] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[2,3,5,4,1] => [2,5,4,3,1] => 0000 => ? = 3 + 1
[2,4,1,3,5] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[2,4,1,5,3] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[2,4,3,1,5] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[2,4,3,5,1] => [2,5,4,3,1] => 0000 => ? = 3 + 1
[2,4,5,1,3] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[2,4,5,3,1] => [2,5,4,3,1] => 0000 => ? = 3 + 1
[2,5,1,3,4] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[2,5,1,4,3] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[2,5,3,1,4] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[2,5,3,4,1] => [2,5,4,3,1] => 0000 => ? = 3 + 1
[2,5,4,1,3] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[2,5,4,3,1] => [2,5,4,3,1] => 0000 => ? = 3 + 1
[3,1,2,4,5] => [3,1,5,4,2] => 0000 => ? = 0 + 1
[3,1,2,5,4] => [3,1,5,4,2] => 0000 => ? = 0 + 1
[3,1,4,2,5] => [3,1,5,4,2] => 0000 => ? = 0 + 1
[3,1,4,5,2] => [3,1,5,4,2] => 0000 => ? = 0 + 1
[3,1,5,2,4] => [3,1,5,4,2] => 0000 => ? = 0 + 1
[3,1,5,4,2] => [3,1,5,4,2] => 0000 => ? = 0 + 1
[3,2,1,4,5] => [3,2,1,5,4] => 0010 => 1 = 0 + 1
[3,2,1,5,4] => [3,2,1,5,4] => 0010 => 1 = 0 + 1
[3,2,4,1,5] => [3,2,5,1,4] => 0000 => ? = 1 + 1
[3,2,4,5,1] => [3,2,5,4,1] => 0000 => ? = 2 + 1
[3,2,5,1,4] => [3,2,5,1,4] => 0000 => ? = 1 + 1
[3,2,5,4,1] => [3,2,5,4,1] => 0000 => ? = 2 + 1
[3,4,1,2,5] => [3,5,1,4,2] => 0000 => ? = 0 + 1
[3,4,1,5,2] => [3,5,1,4,2] => 0000 => ? = 0 + 1
[3,4,2,1,5] => [3,5,2,1,4] => 0000 => ? = 0 + 1
[4,3,2,1,5] => [4,3,2,1,5] => 0001 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000264
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Values
[1,2] => [1,2] => [1,2] => ([],2)
 => ? = 0 + 3
[2,1] => [2,1] => [1,2] => ([],2)
 => ? = 0 + 3
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 3
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 3
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 3
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
 => ? = 1 + 3
[3,1,2] => [3,1,2] => [1,2,3] => ([],3)
 => ? = 0 + 3
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
 => ? = 0 + 3
[1,2,3,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[1,2,4,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[1,3,2,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[2,1,3,4] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[2,3,1,4] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 3
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 3
[2,4,1,3] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 3
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 3
[3,1,2,4] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[3,1,4,2] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 3
[3,4,1,2] => [3,4,1,2] => [1,2,3,4] => ([],4)
 => ? = 0 + 3
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => ([],4)
 => ? = 0 + 3
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => ? = 0 + 3
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => ? = 0 + 3
[4,2,1,3] => [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
 => ? = 0 + 3
[4,2,3,1] => [4,2,3,1] => [1,2,3,4] => ([],4)
 => ? = 1 + 3
[4,3,1,2] => [4,3,1,2] => [1,2,3,4] => ([],4)
 => ? = 0 + 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([],4)
 => ? = 0 + 3
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,3,1,6,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,3,1,6,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,4,1,3,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,4,1,5,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,4,1,6,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,4,1,6,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,5,1,3,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,5,1,4,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,5,1,4,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,5,1,6,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,5,1,6,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,6,1,3,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,6,1,4,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,6,1,4,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[2,6,1,5,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 1 + 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.