Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000698
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => [2]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => [2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => [2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => [2]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [4,3,2,1]
 => [3,2,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [4,3,2,1]
 => [3,2,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [3,3,2,1]
 => [3,2,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [3,3,2,1]
 => [3,2,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [3,3,2,1]
 => [3,2,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [4,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [4,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [4,3,1,1]
 => [3,1,1]
 => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [4,3,1,1]
 => [3,1,1]
 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [3,3,1,1]
 => [3,1,1]
 => 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [3,3,1,1]
 => [3,1,1]
 => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [3,3,1,1]
 => [3,1,1]
 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [4,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [4,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [4,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [4,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001928
(load all 2 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 29%
Values
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
 => [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
 => [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
 => [13,11,12,8,9,10,5,6,7,1,2,3,4] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
 => [12,10,11,7,8,9,4,5,6,1,2,3] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
 => [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
 => [13,11,12,9,10,5,6,7,8,1,2,3,4] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
 => [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
 => [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,4,5,6,1,2,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
 => [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,5,6,1,2,3,4] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,4,5,1,2,3] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
 => [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
 => [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
 => [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
 => [12,11,8,9,10,5,6,7,1,2,3,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
 => [11,10,7,8,9,4,5,6,1,2,3] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
 => [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
 => [12,11,9,10,5,6,7,8,1,2,3,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
 => [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
 => [11,10,8,9,5,6,7,1,2,3,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [[1,2,3],[4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,4,5,6,1,2,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
 => [11,10,8,9,6,7,1,2,3,4,5] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9],[10]]
 => [10,9,7,8,5,6,1,2,3,4] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [9,8,6,7,4,5,1,2,3] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
 => [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
 => [11,10,9,5,6,7,8,1,2,3,4] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
 => [11,10,9,6,7,8,1,2,3,4,5] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9],[10]]
 => [10,9,8,5,6,7,1,2,3,4] => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10]]
 => [10,9,8,6,7,1,2,3,4,5] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [9,8,7,5,6,1,2,3,4] => ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ? = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 2 = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 3 = 2 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 2 = 1 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 2 = 1 + 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 2 = 1 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2 = 1 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2 = 1 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 1 + 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 3 = 2 + 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 2 = 1 + 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 2 = 1 + 1
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 2 = 1 + 1
[1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 2 = 1 + 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2 = 1 + 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2 = 1 + 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 1 + 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 3 = 2 + 1
[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 2 = 1 + 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 2 = 1 + 1
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 2 = 1 + 1
[1,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 2 = 1 + 1
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2 = 1 + 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 2 = 1 + 1
[1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 2 = 1 + 1
[1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 3 = 2 + 1
[1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2 = 1 + 1
[1,1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 2 = 1 + 1
Description
The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000689
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 0 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 0 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [7,7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [3,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [5,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => ? = 0 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [8,3,3]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [9,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [7,4,1,1]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [3,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [6,3,3,1]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1,1]
 => [5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => [4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => [6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,1,1]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [5,2,1,1]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [3,3,3]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,1,1]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,1]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
 => [3,3,3]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1,1,1]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => [3,3,3]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
 => [3,3,3]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$. This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid. An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].