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Your data matches 25 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000645
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> 6
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St001232
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000957
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 69%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 69%
Values
[1,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 5
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 5
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,1,4] => ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ? = 9
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 12
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,1,2,4] => ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,1,2,5] => ? = 10
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 10
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,1,3,4] => ? = 10
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [5,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [6,7,4,2,1,3,5] => ? = 7
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,1,3,6] => ? = 8
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,1,4,5] => ? = 10
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,1,4,6] => ? = 7
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => ? = 8
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 12
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 7
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [6,3,3]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,5,1,2,3,6] => ? = 9
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,7,3,1,2,4,5] => ? = 8
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,2,4,6] => ? = 6
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 8
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => ? = 9
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => ? = 6
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => ? = 6
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 7
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 10
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => ? = 6
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 6
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 6
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000081
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 77%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 77%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> ([],2)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5] => ([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,1,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2,3,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 12
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,1,7,6,5] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(6,3),(6,4),(6,5)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,2,5,4,1,7,6] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,2,3,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,2,3,5,1,7,6] => ([(0,6),(1,4),(1,5),(2,3),(3,6),(6,4),(6,5)],7)
=> ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,2,3,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [5,2,4,3,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [5,2,4,3,6,1,7] => ([(0,6),(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,2,4,3,7,6,1] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => ([(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 12
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,3,2,5,1,7,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,3,2,6,5,1,7] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 9
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,3,2,4,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,3,2,4,7,6,1] => ([(1,4),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,3,2,5,4,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,2,5,4,7,1] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [7,3,2,6,5,4,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,3,4,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,3,4,2,7,6,1] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [7,3,4,2,6,5,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,3,5,4,2,7,1] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [7,3,5,4,2,6,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => ([(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 5
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [7,4,3,2,6,5,1] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 5
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [7,4,3,5,2,6,1] => ([(2,6),(3,5),(4,5),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [7,4,3,6,5,2,1] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,5,3,4,2,7,1] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [7,5,3,4,2,6,1] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 4
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [7,5,3,4,6,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 3
Description
The number of edges of a graph.
Matching statistic: St001584
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 77%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 77%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 9
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 12
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 10
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> ? = 10
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 8
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 10
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 7
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 8
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 7
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 10
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 7
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 8
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 8
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 12
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 7
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 9
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 9
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 8
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 6
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 8
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 7
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 9
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 6
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 7
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 6
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 7
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 6
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 5
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 10
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 6
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 8
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 6
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 5
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 6
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 6
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5
Description
The area statistic between a Dyck path and its bounce path.
The bounce path [[Mp00099]] is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.
Matching statistic: St001869
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001869: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 77%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001869: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 77%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> ([],2)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5] => ([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,1,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2,3,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 12
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,1,7,6,5] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(6,3),(6,4),(6,5)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,2,5,4,1,7,6] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,2,3,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,2,3,5,1,7,6] => ([(0,6),(1,4),(1,5),(2,3),(3,6),(6,4),(6,5)],7)
=> ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [4,2,3,6,5,1,7] => ([(0,6),(1,4),(1,5),(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,2,3,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [5,2,4,3,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [5,2,4,3,6,1,7] => ([(0,6),(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,2,4,3,7,6,1] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => ([(1,6),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => ([(2,3),(2,4),(2,5),(2,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 12
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,3,2,5,1,7,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,3,2,6,5,1,7] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 9
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,3,2,4,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(4,5),(4,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [5,3,2,4,6,1,7] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,3,2,4,7,6,1] => ([(1,4),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,3,2,5,4,1,7] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,2,5,4,7,1] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [7,3,2,6,5,4,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,3,4,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 9
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,3,4,2,6,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,3,4,2,7,6,1] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 7
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,4,2,5,1,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [7,3,4,2,6,5,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 7
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,3,5,4,2,7,1] => ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 6
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [7,3,5,4,2,6,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 5
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ([(3,4),(3,5),(3,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 6
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 8
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,4,3,2,5,1,7] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 6
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => ([(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? = 5
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [7,4,3,2,6,5,1] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 6
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 6
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 5
Description
The maximum cut size of a graph.
A '''cut''' is a set of edges which connect different sides of a vertex partition $V = A \sqcup B$.
Matching statistic: St000327
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 54%
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 54%
Values
[1,0]
=> [1] => ([],1)
=> ? = 0
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([],5)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? = 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5] => ([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,1,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2,3,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ? = 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 5
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 4
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ? = 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(4,5)],6)
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([],6)
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 9
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 12
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,1,7,6,5] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(6,3),(6,4),(6,5)],7)
=> ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,2,5,4,1,7,6] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 8
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,2,3,1,7,6,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 10
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,2,3,5,1,7,6] => ([(0,6),(1,4),(1,5),(2,3),(3,6),(6,4),(6,5)],7)
=> ? = 7
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [4,2,3,6,5,1,7] => ([(0,6),(1,4),(1,5),(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,2,3,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 7
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [5,2,4,3,1,7,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 10
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
Matching statistic: St001880
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 31%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 31%
Values
[1,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,0,1,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 5
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 4
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[3,3,3,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[4,4,1,1],[]]
=> ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
=> ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ? = 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[3,3,3],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[4,4,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
=> ? = 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [[5,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ? = 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ? = 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [[5,3,1],[]]
=> ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
=> ? = 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ? = 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ? = 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ? = 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ? = 5
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? = 4
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[2,2,2,2,2],[]]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> [[3,3,3,3,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1]
=> [[4,4,4,1,1],[]]
=> ?
=> ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1]
=> [[5,5,1,1,1],[]]
=> ?
=> ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10)
=> ? = 9
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [[3,3,3,3],[]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 12
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001681
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001681: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 38%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001681: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 4 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 4 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 8 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => ([(0,5),(1,5),(2,3),(3,4)],6)
=> ([(0,4),(0,5),(0,6),(1,14),(2,7),(2,8),(3,2),(3,11),(3,12),(4,10),(4,16),(5,10),(5,15),(6,3),(6,15),(6,16),(7,17),(8,17),(10,1),(10,18),(11,7),(11,19),(12,8),(12,19),(13,9),(14,13),(15,11),(15,18),(16,12),(16,18),(17,9),(18,14),(18,19),(19,13),(19,17)],20)
=> ? = 7 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ?
=> ? = 8 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,9),(2,7),(2,8),(3,2),(3,11),(3,12),(4,10),(4,13),(5,10),(5,14),(6,3),(6,13),(6,14),(7,17),(8,1),(8,17),(10,15),(11,7),(11,16),(12,8),(12,16),(13,11),(13,15),(14,12),(14,15),(15,16),(16,17),(17,9)],18)
=> ? = 7 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 9 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ?
=> ? = 6 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,8),(2,10),(2,12),(3,9),(3,11),(4,13),(4,14),(5,3),(5,13),(5,15),(6,2),(6,14),(6,15),(7,1),(7,19),(9,17),(10,18),(11,7),(11,17),(12,7),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,8)],20)
=> ? = 7 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 6 + 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => ([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,15),(2,11),(3,7),(3,8),(4,10),(4,14),(5,10),(5,13),(6,3),(6,13),(6,14),(7,17),(8,17),(10,1),(10,16),(11,9),(12,9),(13,7),(13,16),(14,8),(14,16),(15,11),(15,12),(16,15),(16,17),(17,12)],18)
=> ? = 7 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ? = 5 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 5 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,11),(2,7),(3,8),(3,9),(4,10),(4,12),(5,10),(5,13),(6,3),(6,12),(6,13),(8,15),(9,1),(9,15),(10,14),(11,7),(12,8),(12,14),(13,9),(13,14),(14,15),(15,11)],16)
=> ? = 6 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 5 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 3 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 8 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,9),(2,7),(2,8),(3,2),(3,11),(3,12),(4,10),(4,13),(5,10),(5,14),(6,3),(6,13),(6,14),(7,17),(8,1),(8,17),(10,15),(11,7),(11,16),(12,8),(12,16),(13,11),(13,15),(14,12),(14,15),(15,16),(16,17),(17,9)],18)
=> ? = 5 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 6 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,11),(2,7),(3,8),(3,9),(4,10),(4,12),(5,10),(5,13),(6,3),(6,12),(6,13),(8,15),(9,1),(9,15),(10,14),(11,7),(12,8),(12,14),(13,9),(13,14),(14,15),(15,11)],16)
=> ? = 5 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 4 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(1,11),(2,7),(2,8),(3,8),(3,9),(4,7),(4,9),(5,1),(5,12),(6,5),(6,10),(7,13),(8,13),(9,6),(9,13),(10,12),(12,11),(13,10)],14)
=> ? = 5 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 4 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 3 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3 = 2 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,6),(0,7),(1,11),(2,5),(2,15),(3,13),(4,3),(4,17),(5,4),(5,16),(6,2),(6,14),(7,1),(7,14),(9,12),(10,9),(11,10),(12,8),(13,8),(14,11),(14,15),(15,10),(15,16),(16,9),(16,17),(17,12),(17,13)],18)
=> ? = 10 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ?
=> ? = 9 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,1,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 11 + 1
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ?
=> ? = 11 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ?
=> ? = 9 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 5 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? = 12 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,1,2,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 8 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 10 + 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 10 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 0 + 1
Description
The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element.
For example, the pentagon lattice has three such sets: the bottom element, and the two antichains of size two. The cube is the smallest lattice which has such sets of three different sizes: the bottom element, six antichains of size two and one antichain of size three.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 38%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [1] => ([],1)
=> ([(0,1)],2)
=> ? = 0
[1,0,1,0]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? = 5
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 5
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? = 4
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 8
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => ([(0,5),(1,5),(2,3),(3,4)],6)
=> ([(0,4),(0,5),(0,6),(1,14),(2,7),(2,8),(3,2),(3,11),(3,12),(4,10),(4,16),(5,10),(5,15),(6,3),(6,15),(6,16),(7,17),(8,17),(10,1),(10,18),(11,7),(11,19),(12,8),(12,19),(13,9),(14,13),(15,11),(15,18),(16,12),(16,18),(17,9),(18,14),(18,19),(19,13),(19,17)],20)
=> ? = 7
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ?
=> ? = 8
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,9),(2,7),(2,8),(3,2),(3,11),(3,12),(4,10),(4,13),(5,10),(5,14),(6,3),(6,13),(6,14),(7,17),(8,1),(8,17),(10,15),(11,7),(11,16),(12,8),(12,16),(13,11),(13,15),(14,12),(14,15),(15,16),(16,17),(17,9)],18)
=> ? = 7
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 9
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ?
=> ? = 6
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,8),(2,10),(2,12),(3,9),(3,11),(4,13),(4,14),(5,3),(5,13),(5,15),(6,2),(6,14),(6,15),(7,1),(7,19),(9,17),(10,18),(11,7),(11,17),(12,7),(12,18),(13,9),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,17),(16,18),(17,19),(18,19),(19,8)],20)
=> ? = 7
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 6
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => ([(0,5),(1,5),(2,3),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,15),(2,11),(3,7),(3,8),(4,10),(4,14),(5,10),(5,13),(6,3),(6,13),(6,14),(7,17),(8,17),(10,1),(10,16),(11,9),(12,9),(13,7),(13,16),(14,8),(14,16),(15,11),(15,12),(16,15),(16,17),(17,12)],18)
=> ? = 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
=> ?
=> ? = 5
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 5
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,11),(2,7),(3,8),(3,9),(4,10),(4,12),(5,10),(5,13),(6,3),(6,12),(6,13),(8,15),(9,1),(9,15),(10,14),(11,7),(12,8),(12,14),(13,9),(13,14),(14,15),(15,11)],16)
=> ? = 6
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 5
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 8
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,9),(2,7),(2,8),(3,2),(3,11),(3,12),(4,10),(4,13),(5,10),(5,14),(6,3),(6,13),(6,14),(7,17),(8,1),(8,17),(10,15),(11,7),(11,16),(12,8),(12,16),(13,11),(13,15),(14,12),(14,15),(15,16),(16,17),(17,9)],18)
=> ? = 5
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 6
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,11),(2,7),(3,8),(3,9),(4,10),(4,12),(5,10),(5,13),(6,3),(6,12),(6,13),(8,15),(9,1),(9,15),(10,14),(11,7),(12,8),(12,14),(13,9),(13,14),(14,15),(15,11)],16)
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(1,11),(2,7),(2,8),(3,8),(3,9),(4,7),(4,9),(5,1),(5,12),(6,5),(6,10),(7,13),(8,13),(9,6),(9,13),(10,12),(12,11),(13,10)],14)
=> ? = 5
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 4
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ([(0,6),(0,7),(1,11),(2,5),(2,15),(3,13),(4,3),(4,17),(5,4),(5,16),(6,2),(6,14),(7,1),(7,14),(9,12),(10,9),(11,10),(12,8),(13,8),(14,11),(14,15),(15,10),(15,16),(16,9),(16,17),(17,12),(17,13)],18)
=> ? = 10
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ?
=> ? = 9
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,1,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 11
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,1,5] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ?
=> ? = 11
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ([(1,3),(2,6),(3,5),(4,6),(5,4)],7)
=> ?
=> ? = 9
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ? = 5
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ([(0,6),(0,7),(1,4),(1,16),(2,5),(2,15),(3,13),(4,12),(5,3),(5,19),(6,1),(6,17),(7,2),(7,17),(9,11),(10,8),(11,8),(12,9),(13,10),(14,9),(14,18),(15,14),(15,19),(16,12),(16,14),(17,15),(17,16),(18,10),(18,11),(19,13),(19,18)],20)
=> ? = 12
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,1,2,4] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ?
=> ? = 10
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,8),(3,8),(4,6),(5,4),(6,1),(7,5),(8,7)],9)
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St000075The orbit size of a standard tableau under promotion. St000029The depth of a permutation. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000018The number of inversions of a permutation. St000463The number of admissible inversions of a permutation. St001877Number of indecomposable injective modules with projective dimension 2.
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