Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000475
Mapping
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 0
[1,1]
 => 2
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 3
[4]
 => 0
[3,1]
 => 1
[2,2]
 => 0
[2,1,1]
 => 2
[1,1,1,1]
 => 4
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 0
[3,1,1]
 => 2
[2,2,1]
 => 1
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 5
[6]
 => 0
[5,1]
 => 1
[4,2]
 => 0
[4,1,1]
 => 2
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 3
[2,2,2]
 => 0
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 6
[5,2]
 => 0
[4,3]
 => 0
[4,2,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 0
[3,2,1,1]
 => 2
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 3
[5,3]
 => 0
[4,4]
 => 0
[4,3,1]
 => 1
[4,2,2]
 => 0
[3,3,2]
 => 0
[3,3,1,1]
 => 2
[3,2,2,1]
 => 1
[2,2,2,2]
 => 0
[2,2,2,1,1]
 => 2
[5,4]
 => 0
[4,4,1]
 => 1
[4,3,2]
 => 0
[3,3,3]
 => 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000241
(load all 5 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1] => 1
[2]
 => [1,0,1,0]
 => [1,2] => 0
[1,1]
 => [1,1,0,0]
 => [2,1] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => 0
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => 3
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => 0
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => 2
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,4] => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,4,5,6,2,3] => 0
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,2,3,5] => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
Description
The number of cyclical small excedances. A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000025
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 5 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 7 = 6 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4 = 3 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 3 + 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 5 + 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 3 + 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 4 + 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 8 = 6 + 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 1 + 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 5 = 3 + 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4 = 2 + 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 1 + 2
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 0 + 2
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 4 = 2 + 2
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 0 + 2
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000022
(load all 6 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1] => 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,5,6] => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,2,4,5,6] => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,6,4] => 0
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,5,3,6] => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => 0
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [3,1,4,2,5,6] => 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6] => 3
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => 0
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,4,5,2,6] => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,1,2,6,3,4] => 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,1,5,6] => 2
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,1,5,2,3,6] => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [3,4,1,2,5,6] => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => 0
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,1,6] => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,5,6,2,4] => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000117
(load all 3 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,2]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 3
[2,2,2]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,2]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 0
[4,3]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 0
[4,2,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 1
[3,3,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[3,2,2]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 0
[3,2,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[2,2,2,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 3
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 0
[4,4]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1
[4,2,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 0
[3,3,2]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0
[3,3,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 2
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 1
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 2
[5,4]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 0
[4,4,1]
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1
[4,3,2]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 0
[3,3,3]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
Description
The number of centered tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.
Matching statistic: St000221
(load all 15 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1] => 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => 0
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 0
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [3,2,4,1,5,6] => 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6] => 3
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => 0
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [3,4,2,1,5,6] => 2
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,2,4,5,1,6] => 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,1,5,6] => 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,4,6,3,2,1] => 0
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,3,5,2,6,1] => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 0
Description
The number of strong fixed points of a permutation. $i$ is called a strong fixed point of $\pi$ if 1. $j < i$ implies $\pi_j < \pi_i$, and 2. $j > i$ implies $\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St000297
(load all 4 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 01 => 10 => 1
[2]
 => 100 => 001 => 010 => 0
[1,1]
 => 110 => 011 => 110 => 2
[3]
 => 1000 => 0001 => 0010 => 0
[2,1]
 => 1010 => 0101 => 1010 => 1
[1,1,1]
 => 1110 => 0111 => 1110 => 3
[4]
 => 10000 => 00001 => 00010 => 0
[3,1]
 => 10010 => 01001 => 10010 => 1
[2,2]
 => 1100 => 0011 => 0110 => 0
[2,1,1]
 => 10110 => 01101 => 11010 => 2
[1,1,1,1]
 => 11110 => 01111 => 11110 => 4
[5]
 => 100000 => 000001 => 000010 => 0
[4,1]
 => 100010 => 010001 => 100010 => 1
[3,2]
 => 10100 => 00101 => 01010 => 0
[3,1,1]
 => 100110 => 011001 => 110010 => 2
[2,2,1]
 => 11010 => 01011 => 10110 => 1
[2,1,1,1]
 => 101110 => 011101 => 111010 => 3
[1,1,1,1,1]
 => 111110 => 011111 => 111110 => 5
[6]
 => 1000000 => 0000001 => 0000010 => 0
[5,1]
 => 1000010 => 0100001 => 1000010 => 1
[4,2]
 => 100100 => 001001 => 010010 => 0
[4,1,1]
 => 1000110 => 0110001 => 1100010 => 2
[3,3]
 => 11000 => 00011 => 00110 => 0
[3,2,1]
 => 101010 => 010101 => 101010 => 1
[3,1,1,1]
 => 1001110 => 0111001 => 1110010 => 3
[2,2,2]
 => 11100 => 00111 => 01110 => 0
[2,2,1,1]
 => 110110 => 011011 => 110110 => 2
[2,1,1,1,1]
 => 1011110 => 0111101 => 1111010 => 4
[1,1,1,1,1,1]
 => 1111110 => 0111111 => 1111110 => 6
[5,2]
 => 1000100 => 0010001 => 0100010 => 0
[4,3]
 => 101000 => 000101 => 001010 => 0
[4,2,1]
 => 1001010 => 0101001 => 1010010 => 1
[3,3,1]
 => 110010 => 010011 => 100110 => 1
[3,2,2]
 => 101100 => 001101 => 011010 => 0
[3,2,1,1]
 => 1010110 => 0110101 => 1101010 => 2
[2,2,2,1]
 => 111010 => 010111 => 101110 => 1
[2,2,1,1,1]
 => 1101110 => 0111011 => 1110110 => 3
[5,3]
 => 1001000 => 0001001 => 0010010 => 0
[4,4]
 => 110000 => 000011 => 000110 => 0
[4,3,1]
 => 1010010 => 0100101 => 1001010 => 1
[4,2,2]
 => 1001100 => 0011001 => 0110010 => 0
[3,3,2]
 => 110100 => 001011 => 010110 => 0
[3,3,1,1]
 => 1100110 => 0110011 => 1100110 => 2
[3,2,2,1]
 => 1011010 => 0101101 => 1011010 => 1
[2,2,2,2]
 => 111100 => 001111 => 011110 => 0
[2,2,2,1,1]
 => 1110110 => 0110111 => 1101110 => 2
[5,4]
 => 1010000 => 0000101 => 0001010 => 0
[4,4,1]
 => 1100010 => 0100011 => 1000110 => 1
[4,3,2]
 => 1010100 => 0010101 => 0101010 => 0
[3,3,3]
 => 111000 => 000111 => 001110 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000895
(load all 3 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0]
 => [[1]]
 => 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 2
[3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 0
[2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 0
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 1
[2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 0
[4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 1
[3,2]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => 0
[3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 2
[2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 0
[5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 1
[4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 0
[4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 1
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 3
[2,2,2]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => 0
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 2
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 6
[5,2]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 0
[4,3]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 0
[4,2,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 1
[3,3,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[3,2,2]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 0
[3,2,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 2
[2,2,2,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => 3
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 0
[4,4]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 1
[4,2,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 0
[3,3,2]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[3,3,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => 2
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 1
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[2,2,2,1,1]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => 2
[5,4]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 0
[4,4,1]
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => 1
[4,3,2]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => 0
[3,3,3]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St001008
(load all 10 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[3,2]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
[2,2,2]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,2]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0
[4,3]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[4,2,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
[3,3,1]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,2,2]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[3,2,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2
[2,2,2,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 3
[5,3]
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0
[4,4]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[4,3,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1
[4,2,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0
[3,3,2]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
[3,3,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 2
[3,2,2,1]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1
[2,2,2,2]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,2,2,1,1]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 2
[5,4]
 => [2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0
[4,4,1]
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[4,3,2]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 0
[3,3,3]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
Description
Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000971The smallest closer of a set partition. St000674The number of hills of a Dyck path. St000873The aix statistic of a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St000061The number of nodes on the left branch of a binary tree. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000215The number of adjacencies of a permutation, zero appended. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph.