Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000420
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 5
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 14
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 9
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 42
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 28
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 28
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 19
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 28
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 19
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 9
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 12
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
St000419: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 41 = 42 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 18 = 19 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 18 = 19 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 8 = 9 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 11 = 12 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Mp00254: Permutations Inverse fireworks mapPermutations
St000280: Permutations ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 92%
Values
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 5
[1,3,2] => [1,3,2] => 3
[2,1,3] => [2,1,3] => 2
[2,3,1] => [1,3,2] => 3
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 14
[1,2,4,3] => [1,2,4,3] => 9
[1,3,2,4] => [1,3,2,4] => 7
[1,3,4,2] => [1,2,4,3] => 9
[1,4,2,3] => [1,4,2,3] => 7
[1,4,3,2] => [1,4,3,2] => 4
[2,1,3,4] => [2,1,3,4] => 5
[2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [1,3,2,4] => 7
[2,3,4,1] => [1,2,4,3] => 9
[2,4,1,3] => [2,4,1,3] => 7
[2,4,3,1] => [1,4,3,2] => 4
[3,1,2,4] => [3,1,2,4] => 5
[3,1,4,2] => [2,1,4,3] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => 3
[3,4,1,2] => [2,4,1,3] => 7
[3,4,2,1] => [1,4,3,2] => 4
[4,1,2,3] => [4,1,2,3] => 5
[4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,1,3,2] => 3
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 42
[1,2,3,5,4] => [1,2,3,5,4] => 28
[1,2,4,3,5] => [1,2,4,3,5] => 23
[1,2,4,5,3] => [1,2,3,5,4] => 28
[1,2,5,3,4] => [1,2,5,3,4] => 23
[1,2,5,4,3] => [1,2,5,4,3] => 14
[1,3,2,4,5] => [1,3,2,4,5] => 19
[1,3,2,5,4] => [1,3,2,5,4] => 12
[1,3,4,2,5] => [1,2,4,3,5] => 23
[1,3,4,5,2] => [1,2,3,5,4] => 28
[1,3,5,2,4] => [1,3,5,2,4] => 23
[1,3,5,4,2] => [1,2,5,4,3] => 14
[1,4,2,3,5] => [1,4,2,3,5] => 19
[1,4,2,5,3] => [1,3,2,5,4] => 12
[1,4,3,2,5] => [1,4,3,2,5] => 9
[1,4,3,5,2] => [1,3,2,5,4] => 12
[1,4,5,2,3] => [1,3,5,2,4] => 23
[1,4,5,3,2] => [1,2,5,4,3] => 14
[1,4,7,2,3,5,6] => [1,4,7,2,3,5,6] => ? = 202
[1,4,7,2,3,6,5] => [1,4,7,2,3,6,5] => ? = 136
[1,4,7,2,6,3,5] => [1,4,7,2,6,3,5] => ? = 113
[1,4,7,3,2,5,6] => [1,4,7,3,2,5,6] => ? = 95
[1,4,7,3,2,6,5] => [1,4,7,3,2,6,5] => ? = 61
[1,4,7,3,6,2,5] => [1,4,7,3,6,2,5] => ? = 113
[1,4,7,6,2,3,5] => [1,4,7,6,2,3,5] => ? = 95
[1,4,7,6,3,2,5] => [1,4,7,6,3,2,5] => ? = 47
[1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 174
[1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 118
[1,5,2,3,7,4,6] => [1,5,2,3,7,4,6] => ? = 99
[1,5,2,4,3,6,7] => [1,5,2,4,3,6,7] => ? = 85
[1,5,2,4,3,7,6] => [1,5,2,4,3,7,6] => ? = 55
[1,5,2,4,7,3,6] => [1,5,2,4,7,3,6] => ? = 99
[1,5,2,7,3,4,6] => [1,5,2,7,3,4,6] => ? = 85
[1,5,2,7,4,3,6] => [1,5,2,7,4,3,6] => ? = 43
[1,5,3,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 70
[1,5,3,2,4,7,6] => [1,5,3,2,4,7,6] => ? = 46
[1,5,3,2,7,4,6] => [1,5,3,2,7,4,6] => ? = 37
[1,5,3,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 85
[1,5,3,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 55
[1,5,3,4,7,2,6] => [1,5,2,4,7,3,6] => ? = 99
[1,5,3,7,2,4,6] => [1,5,3,7,2,4,6] => ? = 85
[1,5,3,7,4,2,6] => [1,5,2,7,4,3,6] => ? = 43
[1,5,4,2,3,6,7] => [1,5,4,2,3,6,7] => ? = 70
[1,5,4,2,3,7,6] => [1,5,4,2,3,7,6] => ? = 46
[1,5,4,2,7,3,6] => [1,5,4,2,7,3,6] => ? = 37
[1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 29
[1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 18
[1,5,4,3,7,2,6] => [1,5,4,3,7,2,6] => ? = 37
[1,5,4,7,2,3,6] => [1,5,4,7,2,3,6] => ? = 85
[1,5,4,7,3,2,6] => [1,5,4,7,3,2,6] => ? = 43
[1,5,7,2,3,4,6] => [1,5,7,2,3,4,6] => ? = 202
[1,5,7,2,3,6,4] => [1,4,7,2,3,6,5] => ? = 136
[1,5,7,2,4,3,6] => [1,5,7,2,4,3,6] => ? = 113
[1,5,7,2,4,6,3] => [1,4,7,2,3,6,5] => ? = 136
[1,5,7,2,6,3,4] => [1,4,7,2,6,3,5] => ? = 113
[1,5,7,3,2,4,6] => [1,5,7,3,2,4,6] => ? = 95
[1,5,7,3,2,6,4] => [1,4,7,3,2,6,5] => ? = 61
[1,5,7,3,4,2,6] => [1,5,7,2,4,3,6] => ? = 113
[1,5,7,3,4,6,2] => [1,4,7,2,3,6,5] => ? = 136
[1,5,7,3,6,2,4] => [1,4,7,3,6,2,5] => ? = 113
[1,5,7,4,2,3,6] => [1,5,7,4,2,3,6] => ? = 95
[1,5,7,4,2,6,3] => [1,4,7,3,2,6,5] => ? = 61
[1,5,7,4,3,2,6] => [1,5,7,4,3,2,6] => ? = 47
[1,5,7,4,3,6,2] => [1,4,7,3,2,6,5] => ? = 61
[1,5,7,4,6,2,3] => [1,4,7,3,6,2,5] => ? = 113
[1,5,7,6,2,3,4] => [1,4,7,6,2,3,5] => ? = 95
[1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => ? = 47
[1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 174
Description
The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations.