searching the database
Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001232
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to 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]
=> [1,0,1,0]
=> 1
[1,2] => [2] => [2]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [2] => [2]
=> [1,1,0,0,1,0]
=> 1
[1,2,3] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,3] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,3,1] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,4,3] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,3,4,1] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,4,1,3] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,1,4,2] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,3,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,3,1,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,3,4,1,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,3,4,5,1] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,3,5,1,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,4,1,3,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,4,5,1,3] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,5,1,3,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[3,1,2,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[3,4,1,2,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[3,4,5,1,2] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[3,5,1,2,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,1,2,3,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,3,5,2,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,5,1,2,3] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,5,3,2,1] => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,1,2,3,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,3,1,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,3,2,4,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,3,4,2,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,4,1,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,4,2,3,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001000
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1]
=> [1,0,1,0]
=> 1
[1,2] => [2] => [2]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [2] => [2]
=> [1,1,0,0,1,0]
=> 1
[1,2,3] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,3] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,3,1] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2] => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,4,3] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,3,4,1] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,4,1,3] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,1,4,2] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,3,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,3,1,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,1,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,5,1] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,3,5,1,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,4,1,3,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,4,5,1,3] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[2,5,1,3,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[3,1,2,4,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[3,4,1,2,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[3,4,5,1,2] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[3,5,1,2,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[4,1,2,3,5] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[4,3,5,2,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,5,1,2,3] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[4,5,3,2,1] => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,1,2,3,4] => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[5,3,1,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,3,2,4,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,3,4,2,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,4,1,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,4,2,3,1] => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,2,3,4,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,2,3,5,4,6] => [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,3,5,6,4] => [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,3,6,4,5] => [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,3,6,5] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,6,4] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3,6] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,6,3] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,6,4,3,5] => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,4,6,5,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,3,5,6,4,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,3,6,2,5,4] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,3,6,4,5,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,4,5,6,3,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,4,6,2,5,3] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,4,6,3,5,2] => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,5,4,2,3,6] => [1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,5,4,6,2,3] => [1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[2,1,3,4,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,1,4,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,1,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,5,1,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,5,6,1] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,4,6,1,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,5,1,4,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,5,6,1,4] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,3,6,1,4,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,4,1,3,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,4,5,1,3,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,4,5,6,1,3] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,4,6,1,3,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,5,1,3,4,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,5,6,1,3,4] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,6,1,3,4,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,1,2,4,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,4,1,2,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,4,5,1,2,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,4,5,6,1,2] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,4,6,1,2,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,5,1,2,4,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,5,6,1,2,4] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,6,1,2,4,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[4,1,2,3,5,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[4,5,1,2,3,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[4,5,6,1,2,3] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[4,6,1,2,3,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[5,1,2,3,4,6] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[5,6,1,2,3,4] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[6,1,2,3,4,5] => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,3,4,5,7,6,2] => [1,5,1] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 7
[1,3,4,6,7,5,2] => [1,5,1] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 7
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000298
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,4,1,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 3
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 5
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 5
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[2,1,3,6,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[2,1,5,3,6,4] => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[2,1,5,4,3,6] => [5,4,2,1,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000632
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[5,4,1,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 - 1
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 - 1
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 3 - 1
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5 - 1
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5 - 1
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 5 - 1
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5 - 1
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5 - 1
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5 - 1
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5 - 1
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5 - 1
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5 - 1
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 5 - 1
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[2,1,3,6,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[2,1,5,3,6,4] => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[2,1,5,4,3,6] => [5,4,2,1,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 - 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000640: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000640: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1,0]
=> ([],1)
=> ? = 1
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,4,1,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 3
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 5
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 5
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[2,1,3,6,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[2,1,5,3,6,4] => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The rank of the largest boolean interval in a poset.
Matching statistic: St000307
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
[5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3
[5,4,1,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3
[5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 3
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 5
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 5
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[2,1,3,6,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
[2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[2,1,5,3,6,4] => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3
[2,1,5,4,3,6] => [5,4,2,1,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001095
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,4,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,4,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,4,2] => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[3,4,1,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3 - 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[5,3,2,4,1] => [3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 3 - 1
[5,3,4,2,1] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3 - 1
[5,4,1,3,2] => [5,1,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[5,4,2,3,1] => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,5,4,3,2,6] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2 - 1
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 3 - 1
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 3 - 1
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 3 - 1
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 3 - 1
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,4,2,6,5,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ? = 3 - 1
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,5,4,6,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 5 - 1
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,6,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 5 - 1
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 5 - 1
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ? = 5 - 1
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,4,6,2,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ? = 5 - 1
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 5 - 1
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,17),(1,18),(2,12),(2,14),(2,18),(2,19),(3,11),(3,14),(3,17),(3,19),(4,10),(4,13),(4,17),(4,18),(4,19),(5,9),(5,13),(5,17),(5,18),(5,19),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,20),(9,21),(9,22),(9,25),(10,20),(10,21),(10,22),(10,25),(11,15),(11,20),(11,21),(12,15),(12,20),(12,22),(13,16),(13,25),(14,15),(14,25),(15,24),(16,23),(17,16),(17,21),(17,25),(18,16),(18,22),(18,25),(19,16),(19,20),(19,25),(20,23),(20,24),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 5 - 1
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 5 - 1
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,5,4,6,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 5 - 1
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,5,3,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ? = 5 - 1
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,6,4,5,3,2] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 5 - 1
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ? = 5 - 1
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,3,6,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ? = 5 - 1
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ? = 5 - 1
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ? = 5 - 1
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ? = 5 - 1
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,2,6,3,4,5] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 5 - 1
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 5 - 1
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,6,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 5 - 1
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 5 - 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,5,4,3,2,6] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2 - 1
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,5,6,4,3,2] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2 - 1
[2,1,3,6,4,5] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,6,5,3,2,4] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 3 - 1
[2,1,5,3,6,4] => [5,6,2,1,3,4] => [1,5,3,2,6,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 3 - 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The number of non-isomorphic posets with precisely one further covering relation.
Matching statistic: St001964
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => [2,1] => [1,1,0,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 3 - 1
[5,4,1,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 3 - 1
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 - 1
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 - 1
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 3 - 1
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 5 - 1
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5 - 1
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 5 - 1
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5 - 1
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5 - 1
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 5 - 1
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5 - 1
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 5 - 1
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5 - 1
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 5 - 1
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5 - 1
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 5 - 1
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[2,1,3,6,4,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,5,1,3,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,5,6,1,3,4] => [2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001632
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,4,1] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,4,1,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3
[1,4,5,3,2] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3
[1,5,3,4,2] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3
[1,5,4,2,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 3
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3
[5,3,2,4,1] => [3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 3
[5,3,4,2,1] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 3
[5,4,1,3,2] => [5,1,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3
[5,4,2,3,1] => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,5,4,3,2,6] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2
[1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,5,6,4,3,2] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2
[1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,6,5,3,2,4] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 3
[1,2,5,3,6,4] => [5,6,1,2,3,4] => [1,5,3,2,6,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 3
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 3
[1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,5,6,3,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 3
[1,2,6,4,3,5] => [4,6,1,2,3,5] => [1,4,2,6,5,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ? = 3
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,5,4,6,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 5
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,6,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 5
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [1,6,2,3,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 5
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ? = 5
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,4,6,2,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ? = 5
[1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,5,4,6,2,3] => [1,5,4,2,6,3] => [1,2,5,6,3,4] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? = 5
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,17),(1,18),(2,12),(2,14),(2,18),(2,19),(3,11),(3,14),(3,17),(3,19),(4,10),(4,13),(4,17),(4,18),(4,19),(5,9),(5,13),(5,17),(5,18),(5,19),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,20),(9,21),(9,22),(9,25),(10,20),(10,21),(10,22),(10,25),(11,15),(11,20),(11,21),(12,15),(12,20),(12,22),(13,16),(13,25),(14,15),(14,25),(15,24),(16,23),(17,16),(17,21),(17,25),(18,16),(18,22),(18,25),(19,16),(19,20),(19,25),(20,23),(20,24),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 5
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 5
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [1,5,4,6,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 5
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [1,5,3,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ? = 5
[1,6,2,3,5,4] => [6,1,2,5,3,4] => [1,6,4,5,3,2] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 5
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [1,6,3,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ? = 5
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,3,6,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ? = 5
[1,6,4,2,3,5] => [1,4,6,2,3,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ? = 5
[1,6,4,2,5,3] => [4,6,5,1,2,3] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ? = 5
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ? = 5
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,2,6,3,4,5] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 5
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,6,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 5
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [1,2,3,6,4,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5
[1,6,5,2,4,3] => [6,1,5,4,2,3] => [1,6,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 5
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [1,6,2,3,5,4] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 5
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,5,4,3,2,6] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2
[2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,5,6,4,3,2] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? = 2
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,5,1,3,4,6] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,5,6,1,3,4] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,6,1,3,4,5] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001427
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001427: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 30%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001427: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 30%
Values
[1] => [1] => [1] => [-1] => 1
[1,2] => [2,1] => [2,1] => [-1,2] => 1
[2,1] => [1,2] => [1,2] => [2,-1] => 1
[1,2,3] => [2,3,1] => [2,3,1] => [-1,2,3] => 1
[2,1,3] => [1,3,2] => [1,3,2] => [2,-1,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,-1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [3,-1,2] => 1
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => [-1,2,3,4] => 1
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => [-1,2,4,3] => 2
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 1
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 1
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => [3,-1,2,4] => 1
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => [3,-1,4,2] => 2
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [3,2,-1,4] => 2
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [3,2,4,-1] => 2
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [4,-1,2,3] => 1
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 2
[1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => [-1,2,3,4,5] => 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [-1,3,5,4,2] => 3
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => [-1,4,5,3,2] => 3
[1,5,2,4,3] => [3,5,4,2,1] => [3,5,4,2,1] => [-1,5,2,4,3] => 3
[1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => [-1,5,3,4,2] => 3
[1,5,4,2,3] => [4,5,3,2,1] => [4,5,3,2,1] => [-1,5,4,2,3] => 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => [2,-1,3,4,5] => 1
[2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => [2,3,-1,4,5] => 1
[2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => 1
[2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,-1,4] => 1
[2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => [2,4,-1,3,5] => 1
[2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,-1,3] => 1
[2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => [2,5,-1,3,4] => 1
[3,1,2,4,5] => [3,1,4,5,2] => [3,1,4,5,2] => [3,-1,2,4,5] => 1
[3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => [3,4,-1,2,5] => 1
[3,4,5,1,2] => [5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,-1,2] => 1
[3,5,1,2,4] => [4,1,5,2,3] => [4,1,5,2,3] => [3,5,-1,2,4] => 1
[4,1,2,3,5] => [3,4,1,5,2] => [3,4,1,5,2] => [4,-1,2,3,5] => 1
[4,3,5,2,1] => [4,2,1,3,5] => [4,2,1,3,5] => [4,3,5,2,-1] => 3
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,-1,2,3] => 1
[4,5,3,2,1] => [4,3,1,2,5] => [4,3,1,2,5] => [4,5,3,2,-1] => 3
[5,1,2,3,4] => [3,4,5,1,2] => [3,4,5,1,2] => [5,-1,2,3,4] => 1
[5,3,1,4,2] => [5,2,4,1,3] => [5,2,4,1,3] => [5,3,-1,4,2] => 3
[5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => [5,3,2,4,-1] => 3
[5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1,5] => [5,3,4,2,-1] => 3
[5,4,1,3,2] => [5,4,2,1,3] => [5,4,2,1,3] => [5,4,-1,3,2] => 3
[5,4,2,3,1] => [3,4,2,1,5] => [3,4,2,1,5] => [5,4,2,3,-1] => 3
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-1,2,3,4,5,6] => ? = 1
[1,2,3,5,4,6] => [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-1,2,3,5,4,6] => ? = 2
[1,2,3,5,6,4] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-1,2,3,5,6,4] => ? = 2
[1,2,3,6,4,5] => [2,3,5,6,4,1] => [2,3,5,6,4,1] => [-1,2,3,6,4,5] => ? = 2
[1,2,4,3,6,5] => [2,4,3,6,5,1] => [2,4,3,6,5,1] => [-1,2,4,3,6,5] => ? = 3
[1,2,5,3,6,4] => [2,4,6,3,5,1] => [2,4,6,3,5,1] => [-1,2,5,3,6,4] => ? = 3
[1,2,5,4,3,6] => [2,5,4,3,6,1] => [2,5,4,3,6,1] => [-1,2,5,4,3,6] => ? = 3
[1,2,5,4,6,3] => [2,6,4,3,5,1] => [2,6,4,3,5,1] => [-1,2,5,4,6,3] => ? = 3
[1,2,6,4,3,5] => [2,5,4,6,3,1] => [2,5,4,6,3,1] => [-1,2,6,4,3,5] => ? = 3
[1,3,4,6,5,2] => [6,2,3,5,4,1] => [6,2,3,5,4,1] => [-1,3,4,6,5,2] => ? = 5
[1,3,5,6,4,2] => [6,2,5,3,4,1] => [6,2,5,3,4,1] => [-1,3,5,6,4,2] => ? = 5
[1,3,6,2,5,4] => [4,2,6,5,3,1] => [4,2,6,5,3,1] => [-1,3,6,2,5,4] => ? = 5
[1,3,6,4,5,2] => [6,2,4,5,3,1] => [6,2,4,5,3,1] => [-1,3,6,4,5,2] => ? = 5
[1,4,5,6,3,2] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => [-1,4,5,6,3,2] => ? = 5
[1,4,6,2,5,3] => [4,6,2,5,3,1] => [4,6,2,5,3,1] => [-1,4,6,2,5,3] => ? = 5
[1,4,6,3,5,2] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => [-1,4,6,3,5,2] => ? = 5
[1,5,4,2,3,6] => [4,5,3,2,6,1] => [4,5,3,2,6,1] => [-1,5,4,2,3,6] => ? = 5
[1,5,4,6,2,3] => [5,6,3,2,4,1] => [5,6,3,2,4,1] => [-1,5,4,6,2,3] => ? = 5
[1,5,4,6,3,2] => [6,5,3,2,4,1] => [6,5,3,2,4,1] => [-1,5,4,6,3,2] => ? = 5
[1,5,6,2,4,3] => [4,6,5,2,3,1] => [4,6,5,2,3,1] => [-1,5,6,2,4,3] => ? = 5
[1,5,6,3,4,2] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [-1,5,6,3,4,2] => ? = 5
[1,5,6,4,3,2] => [6,5,4,2,3,1] => [6,5,4,2,3,1] => [-1,5,6,4,3,2] => ? = 5
[1,6,2,3,5,4] => [3,4,6,5,2,1] => [3,4,6,5,2,1] => [-1,6,2,3,5,4] => ? = 5
[1,6,2,4,5,3] => [3,6,4,5,2,1] => [3,6,4,5,2,1] => [-1,6,2,4,5,3] => ? = 5
[1,6,3,4,5,2] => [6,3,4,5,2,1] => [6,3,4,5,2,1] => [-1,6,3,4,5,2] => ? = 5
[1,6,4,2,3,5] => [4,5,3,6,2,1] => [4,5,3,6,2,1] => [-1,6,4,2,3,5] => ? = 5
[1,6,4,2,5,3] => [4,6,3,5,2,1] => [4,6,3,5,2,1] => [-1,6,4,2,5,3] => ? = 5
[1,6,4,3,5,2] => [6,4,3,5,2,1] => [6,4,3,5,2,1] => [-1,6,4,3,5,2] => ? = 5
[1,6,4,5,2,3] => [5,6,3,4,2,1] => [5,6,3,4,2,1] => [-1,6,4,5,2,3] => ? = 5
[1,6,4,5,3,2] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => [-1,6,4,5,3,2] => ? = 5
[1,6,5,2,3,4] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => [-1,6,5,2,3,4] => ? = 5
[1,6,5,2,4,3] => [4,6,5,3,2,1] => [4,6,5,3,2,1] => [-1,6,5,2,4,3] => ? = 5
[1,6,5,3,4,2] => [6,4,5,3,2,1] => [6,4,5,3,2,1] => [-1,6,5,3,4,2] => ? = 5
[2,1,3,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => [2,-1,3,4,5,6] => ? = 1
[2,1,3,5,4,6] => [1,3,5,4,6,2] => [1,3,5,4,6,2] => [2,-1,3,5,4,6] => ? = 2
[2,1,3,5,6,4] => [1,3,6,4,5,2] => [1,3,6,4,5,2] => [2,-1,3,5,6,4] => ? = 2
[2,1,3,6,4,5] => [1,3,5,6,4,2] => [1,3,5,6,4,2] => [2,-1,3,6,4,5] => ? = 2
[2,1,4,3,6,5] => [1,4,3,6,5,2] => [1,4,3,6,5,2] => [2,-1,4,3,6,5] => ? = 3
[2,1,5,3,6,4] => [1,4,6,3,5,2] => [1,4,6,3,5,2] => [2,-1,5,3,6,4] => ? = 3
[2,1,5,4,3,6] => [1,5,4,3,6,2] => [1,5,4,3,6,2] => [2,-1,5,4,3,6] => ? = 3
[2,1,5,4,6,3] => [1,6,4,3,5,2] => [1,6,4,3,5,2] => [2,-1,5,4,6,3] => ? = 3
[2,1,6,4,3,5] => [1,5,4,6,3,2] => [1,5,4,6,3,2] => [2,-1,6,4,3,5] => ? = 3
[2,3,1,4,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => [2,3,-1,4,5,6] => ? = 1
[2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,5,4,6,3] => [2,3,-1,5,4,6] => ? = 2
[2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => [2,3,-1,5,6,4] => ? = 2
[2,3,1,6,4,5] => [1,2,5,6,4,3] => [1,2,5,6,4,3] => [2,3,-1,6,4,5] => ? = 2
[2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => [2,3,4,-1,5,6] => ? = 1
[2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,5,-1,6] => ? = 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,-1] => ? = 1
[2,3,4,6,1,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [2,3,4,6,-1,5] => ? = 1
Description
The number of descents of a signed permutation.
A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001960The number of descents of a permutation minus one if its first entry is not one. St000181The number of connected components of the Hasse diagram for the poset. St001896The number of right descents of a signed permutations. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001905The number of preferred parking spots in a parking function less than the index of the car.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!