searching the database
Your data matches 166 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: St001552
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001552: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
[1,4,5,3,2] => 0
Description
The number of inversions between excedances and fixed points of a permutation.
This is,
$$\operatorname{iefp}(\sigma) = \#\{1 \leq i,j \leq n \mid i < j = \sigma(j) < \sigma(i) \}.$$
Matching statistic: St000369
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,6,1,3,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
Matching statistic: St000376
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,6,1,3,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
Description
The bounce deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$$
The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
Matching statistic: St000688
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,6,1,3,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
Description
The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path.
The global dimension is given by [[St000684]] and the dominant dimension is given by [[St000685]]. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least $2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].
Matching statistic: St000691
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> 1 => 0
[2,1] => [1,1,0,0]
=> []
=> ? => ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 01 => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 11 => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0 => 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101 => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 001 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 111 => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 00 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0101 => 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 0001 => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1001 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0001 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0001 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 0111 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1111 => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 0011 => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1011 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0011 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0011 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0111 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1111 => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0111 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1111 => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0111 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0111 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 010 => 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110 => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 000 => 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4}
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000970
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2}
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,6,1,3,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
Matching statistic: St001089
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4}
Description
Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St001167
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4}
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001176
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> [1]
=> 0
[2,1] => [2]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1]
=> 0
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1]
=> 0
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1]
=> 0
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [2]
=> 0
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,3,2,1] => [2,2]
=> [2]
=> [2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
[2,5,6,1,3,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4}
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001185
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%
Values
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,1}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,1}
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1}
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,5,6,1,3,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
The following 156 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001964The interval resolution global dimension of a poset. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000661The number of rises of length 3 of a Dyck path. St000938The number of zeros of the symmetric group character corresponding to the partition. St000934The 2-degree of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000940The number of characters of the symmetric group whose value on the partition is zero. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000941The number of characters of the symmetric group whose value on the partition is even. St001651The Frankl number of a lattice. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001596The number of two-by-two squares inside a skew partition. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000699The toughness times the least common multiple of 1,. St001498The normalised height of a Nakayama algebra with magnitude 1. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000454The largest eigenvalue of a graph if it is integral. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001330The hat guessing number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001571The Cartan determinant of the integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001845The number of join irreducibles minus the rank of a lattice. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001561The value of the elementary symmetric function evaluated at 1. St001487The number of inner corners of a skew partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001846The number of elements which do not have a complement in the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000527The width of the poset. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001867The number of alignments of type EN of a signed permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001927Sparre Andersen's number of positives of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000907The number of maximal antichains of minimal length in a poset.
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!