Processing math: 100%

Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000445
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
 => 1
[1,2] => 1 => [1,1] => [1,0,1,0]
 => 2
[2,1] => 0 => [2] => [1,1,0,0]
 => 0
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000475
(load all 7 compositions to match this statistic)
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1]
 => 1
[1,2] => 1 => [1,1] => [1,1]
 => 2
[2,1] => 0 => [2] => [2]
 => 0
[1,2,3] => 11 => [1,1,1] => [1,1,1]
 => 3
[1,3,2] => 10 => [1,2] => [2,1]
 => 1
[2,1,3] => 01 => [2,1] => [2,1]
 => 1
[2,3,1] => 00 => [3] => [3]
 => 0
[3,1,2] => 00 => [3] => [3]
 => 0
[3,2,1] => 00 => [3] => [3]
 => 0
[1,2,3,4] => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[1,2,4,3] => 110 => [1,1,2] => [2,1,1]
 => 2
[1,3,2,4] => 101 => [1,2,1] => [2,1,1]
 => 2
[1,3,4,2] => 100 => [1,3] => [3,1]
 => 1
[1,4,2,3] => 100 => [1,3] => [3,1]
 => 1
[1,4,3,2] => 100 => [1,3] => [3,1]
 => 1
[2,1,3,4] => 011 => [2,1,1] => [2,1,1]
 => 2
[2,1,4,3] => 010 => [2,2] => [2,2]
 => 0
[2,3,1,4] => 001 => [3,1] => [3,1]
 => 1
[2,3,4,1] => 000 => [4] => [4]
 => 0
[2,4,1,3] => 000 => [4] => [4]
 => 0
[2,4,3,1] => 000 => [4] => [4]
 => 0
[3,1,2,4] => 001 => [3,1] => [3,1]
 => 1
[3,1,4,2] => 000 => [4] => [4]
 => 0
[3,2,1,4] => 001 => [3,1] => [3,1]
 => 1
[3,2,4,1] => 000 => [4] => [4]
 => 0
[3,4,1,2] => 000 => [4] => [4]
 => 0
[3,4,2,1] => 000 => [4] => [4]
 => 0
[4,1,2,3] => 000 => [4] => [4]
 => 0
[4,1,3,2] => 000 => [4] => [4]
 => 0
[4,2,1,3] => 000 => [4] => [4]
 => 0
[4,2,3,1] => 000 => [4] => [4]
 => 0
[4,3,1,2] => 000 => [4] => [4]
 => 0
[4,3,2,1] => 000 => [4] => [4]
 => 0
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
[1,2,3,5,4] => 1110 => [1,1,1,2] => [2,1,1,1]
 => 3
[1,2,4,3,5] => 1101 => [1,1,2,1] => [2,1,1,1]
 => 3
[1,2,4,5,3] => 1100 => [1,1,3] => [3,1,1]
 => 2
[1,2,5,3,4] => 1100 => [1,1,3] => [3,1,1]
 => 2
[1,2,5,4,3] => 1100 => [1,1,3] => [3,1,1]
 => 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [2,1,1,1]
 => 3
[1,3,2,5,4] => 1010 => [1,2,2] => [2,2,1]
 => 1
[1,3,4,2,5] => 1001 => [1,3,1] => [3,1,1]
 => 2
[1,3,4,5,2] => 1000 => [1,4] => [4,1]
 => 1
[1,3,5,2,4] => 1000 => [1,4] => [4,1]
 => 1
[1,3,5,4,2] => 1000 => [1,4] => [4,1]
 => 1
[1,4,2,3,5] => 1001 => [1,3,1] => [3,1,1]
 => 2
[1,4,2,5,3] => 1000 => [1,4] => [4,1]
 => 1
[1,4,3,2,5] => 1001 => [1,3,1] => [3,1,1]
 => 2
[1,4,3,5,2] => 1000 => [1,4] => [4,1]
 => 1
[1,4,5,2,3] => 1000 => [1,4] => [4,1]
 => 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000247
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => {{1}}
 => ? = 1
[1,2] => [1,2] => [1,2] => {{1},{2}}
 => 2
[2,1] => [2,1] => [2,1] => {{1,2}}
 => 0
[1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 3
[1,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
 => 1
[2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
 => 1
[2,3,1] => [3,2,1] => [2,3,1] => {{1,2,3}}
 => 0
[3,1,2] => [3,2,1] => [2,3,1] => {{1,2,3}}
 => 0
[3,2,1] => [3,2,1] => [2,3,1] => {{1,2,3}}
 => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
 => 0
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
 => 1
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
 => 0
[2,4,3,1] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
 => 1
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
 => 1
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[3,4,1,2] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[3,4,2,1] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[4,2,1,3] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[4,2,3,1] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[4,3,1,2] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,8,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,5,6,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,5,6,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,6,5,7,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,5,6,7,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,5,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[6,7,5,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,5,6,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[6,5,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,6,4,5,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,6,4,5,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,6,7,4,5,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,5,4,6,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,4,5,6,3,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,6,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,6,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,6,7,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,8,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,5,6,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,5,6,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,5,6,7,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,5,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[6,7,5,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[6,5,7,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[5,6,7,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,8,4,3,5,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,6,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,6,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,6,7,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,8,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[6,7,8,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,5,4,3,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,5,4,3,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,4,5,3,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,5,3,4,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,8,5,3,4,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,4,3,5,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,7,3,4,5,6,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,6,5,4,3,7,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[8,6,4,3,5,7,2,1] => [8,7,6,5,4,3,2,1] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 0
[7,6,5,4,3,8,2,1] => [8,7,5,4,3,6,2,1] => [4,5,6,7,1,8,2,3] => {{1,2,4,5,7},{3,6,8}}
 => ? = 0
[6,7,5,4,3,8,2,1] => [8,7,5,4,3,6,2,1] => [4,5,6,7,1,8,2,3] => {{1,2,4,5,7},{3,6,8}}
 => ? = 0
[7,5,6,4,3,8,2,1] => [8,7,5,4,3,6,2,1] => [4,5,6,7,1,8,2,3] => {{1,2,4,5,7},{3,6,8}}
 => ? = 0
[5,6,7,4,3,8,2,1] => [8,7,5,4,3,6,2,1] => [4,5,6,7,1,8,2,3] => {{1,2,4,5,7},{3,6,8}}
 => ? = 0
[7,5,4,6,3,8,2,1] => [8,7,5,4,3,6,2,1] => [4,5,6,7,1,8,2,3] => {{1,2,4,5,7},{3,6,8}}
 => ? = 0
Description
The number of singleton blocks of a set partition.
Matching statistic: St000674
(load all 121 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => ? = 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 0
[6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 0
[7,5,6,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 0
[6,5,7,4,8,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 0
[5,6,7,4,8,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[7,6,4,5,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 0
[6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 0
[7,5,4,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 0
[7,4,5,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 0
[6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 0
[5,6,4,7,8,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 0
[5,4,6,7,8,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[4,5,6,7,8,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 0
[8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[6,5,7,8,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[7,6,8,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[8,6,7,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[7,6,8,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
Description
The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St001126
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 1
[1,2] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 2
[2,1] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,8,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,5,6,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,5,6,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,6,5,7,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,5,6,7,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,5,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,5,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,5,6,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,5,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,6,4,5,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,4,5,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,6,7,4,5,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,5,4,6,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,4,5,6,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,5,4,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,5,4,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,5,6,4,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,5,7,4,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[5,6,7,4,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,4,5,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,4,5,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,5,4,6,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,4,5,6,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,5,4,7,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[5,6,4,7,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,4,5,7,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[5,4,6,7,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[4,5,6,7,8,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,6,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,6,7,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,8,5,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,5,6,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,5,6,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,5,6,7,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,5,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,5,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,5,7,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[5,6,7,8,3,4,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,8,4,3,5,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,6,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,6,7,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,6,8,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[6,7,8,3,4,5,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,5,4,3,6,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
Description
Number of simple module that are 1-regular in the corresponding Nakayama algebra.
Matching statistic: St001691
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
 => 1
[1,2] => [1,2] => [1,2] => ([],2)
 => 2
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
 => 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => 3
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,3,1,5,6,7,4] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,5,7,4,6] => [3,2,1,6,7,4,5] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,5,7,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,6,4,7,5] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,6,5,7,4] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,6,7,4,5] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,6,7,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,7,4,5,6] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,7,4,6,5] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,7,5,4,6] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,7,5,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,7,6,4,5] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,1,7,6,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,4,1,6,7,5] => [4,2,3,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,4,1,7,5,6] => [4,2,3,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,3,4,1,7,6,5] => [4,2,3,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,4,1,3,6,7,5] => [3,4,1,2,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,4,1,3,7,5,6] => [3,4,1,2,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,4,1,3,7,6,5] => [3,4,1,2,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,4,3,1,6,7,5] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,4,3,1,7,5,6] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,4,3,1,7,6,5] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,5,6,7,4] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,5,7,4,6] => [3,2,1,6,7,4,5] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,5,7,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,6,4,7,5] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,6,5,7,4] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,6,7,4,5] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,6,7,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,7,4,5,6] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,7,4,6,5] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,7,5,4,6] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,7,5,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,7,6,4,5] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,2,7,6,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,4,2,6,7,5] => [4,2,3,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,4,2,7,5,6] => [4,2,3,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,4,2,7,6,5] => [4,2,3,1,7,6,5] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,5,6,7,4] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,5,7,4,6] => [3,2,1,6,7,4,5] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,5,7,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,6,4,7,5] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,6,5,7,4] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,6,7,4,5] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,6,7,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,7,4,5,6] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,7,4,6,5] => [3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,7,5,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,7,6,4,5] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
Description
The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.
Matching statistic: St000895
(load all 13 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 58%
Values
[1] => [1] => [1,0]
 => [[1]]
 => 1
[1,2] => [1,2] => [1,0,1,0]
 => [[1,0],[0,1]]
 => 2
[2,1] => [2,1] => [1,1,0,0]
 => [[0,1],[1,0]]
 => 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => 0
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 7
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => ? = 5
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 4
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 4
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 4
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 5
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 4
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,5,7,6,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 4
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 4
[1,2,3,6,5,7,4] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,7,4,5,6] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,7,5,4,6] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,7,6,4,5] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 5
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => ? = 3
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 4
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,5,7,3,6] => [1,2,6,4,7,3,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
[1,2,4,6,3,7,5] => [1,2,5,7,3,6,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
[1,2,4,6,5,7,3] => [1,2,7,5,4,6,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,6,7,3,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,7,3,5,6] => [1,2,5,7,3,6,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,7,3,6,5] => [1,2,5,7,3,6,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,7,5,6,3] => [1,2,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,7,6,3,5] => [1,2,6,7,5,3,4] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,4,7,6,5,3] => [1,2,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 4
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
 => ? = 2
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000022
(load all 4 compositions to match this statistic)
Mapping
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [2,1] => [1,2] => 2
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 3
[1,3,2] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,2,3] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 2
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2
[1,3,4,2] => [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,4,2,3] => [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[1,4,3,2] => [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 0
[2,3,1,4] => [4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[2,4,1,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[2,4,3,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[3,1,2,4] => [4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[3,1,4,2] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 0
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 1
[3,2,4,1] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 0
[3,4,1,2] => [3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[3,4,2,1] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,2,1,3] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,2,3,1] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 0
[4,3,1,2] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => [2,1,3,4,5] => 3
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [4,3,5,2,1] => [4,3,5,2,1] => [2,3,1,4,5] => 2
[1,2,5,3,4] => [3,5,4,2,1] => [3,5,4,2,1] => [3,1,2,4,5] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => [3,1,2,4,5] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 3
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => [2,1,4,3,5] => 1
[1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [4,3,2,5,1] => [4,3,2,5,1] => [2,3,4,1,5] => 1
[1,3,5,2,4] => [3,2,5,4,1] => [3,2,5,4,1] => [3,4,1,2,5] => 1
[1,3,5,4,2] => [3,2,4,5,1] => [3,2,5,4,1] => [3,4,1,2,5] => 1
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [2,4,3,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,4,2,3,5] => 2
[1,4,3,5,2] => [3,4,2,5,1] => [3,5,2,4,1] => [3,1,4,2,5] => 1
[1,4,5,2,3] => [4,2,5,3,1] => [4,2,5,3,1] => [2,4,1,3,5] => 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 5
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 3
[1,2,3,5,6,4,7] => [7,5,4,6,3,2,1] => [7,5,4,6,3,2,1] => [1,3,4,2,5,6,7] => ? = 4
[1,2,3,6,4,5,7] => [7,4,6,5,3,2,1] => [7,4,6,5,3,2,1] => [1,4,2,3,5,6,7] => ? = 4
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,4,6,5,3,2,1] => [1,4,2,3,5,6,7] => ? = 4
[1,2,3,6,5,7,4] => [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => ? = 3
[1,2,3,7,5,6,4] => [5,7,4,6,3,2,1] => [5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => ? = 3
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 5
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 3
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => ? = 3
[1,2,4,3,6,7,5] => [6,5,7,3,4,2,1] => [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 2
[1,2,4,3,7,5,6] => [5,7,6,3,4,2,1] => [5,7,6,3,4,2,1] => [3,1,2,5,4,6,7] => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [5,7,6,3,4,2,1] => [3,1,2,5,4,6,7] => ? = 2
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => [1,2,4,5,3,6,7] => ? = 4
[1,2,4,5,3,7,6] => [6,7,4,3,5,2,1] => [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 2
[1,2,4,5,6,3,7] => [7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,3,4,5,2,6,7] => ? = 3
[1,2,4,5,7,3,6] => [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,5,7,6,3] => [5,4,3,6,7,2,1] => [5,4,3,7,6,2,1] => [3,4,5,1,2,6,7] => ? = 2
[1,2,4,6,3,5,7] => [7,4,3,6,5,2,1] => [7,4,3,6,5,2,1] => [1,4,5,2,3,6,7] => ? = 3
[1,2,4,6,3,7,5] => [4,3,6,5,7,2,1] => [4,3,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 2
[1,2,4,6,5,3,7] => [7,4,3,5,6,2,1] => [7,4,3,6,5,2,1] => [1,4,5,2,3,6,7] => ? = 3
[1,2,4,6,5,7,3] => [5,6,4,3,7,2,1] => [5,7,4,3,6,2,1] => [3,1,4,5,2,6,7] => ? = 2
[1,2,4,6,7,3,5] => [6,4,3,7,5,2,1] => [6,4,3,7,5,2,1] => [2,4,5,1,3,6,7] => ? = 2
[1,2,4,6,7,5,3] => [6,4,3,5,7,2,1] => [6,4,3,7,5,2,1] => [2,4,5,1,3,6,7] => ? = 2
[1,2,4,7,3,5,6] => [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 2
[1,2,4,7,3,6,5] => [4,3,7,5,6,2,1] => [4,3,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 2
[1,2,4,7,5,3,6] => [4,3,5,7,6,2,1] => [4,3,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 2
[1,2,4,7,5,6,3] => [5,7,4,3,6,2,1] => [5,7,4,3,6,2,1] => [3,1,4,5,2,6,7] => ? = 2
[1,2,4,7,6,3,5] => [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 2
[1,2,4,7,6,5,3] => [4,3,5,6,7,2,1] => [4,3,7,6,5,2,1] => [4,5,1,2,3,6,7] => ? = 2
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => [1,2,5,3,4,6,7] => ? = 4
[1,2,5,3,4,7,6] => [6,7,3,5,4,2,1] => [6,7,3,5,4,2,1] => [2,1,5,3,4,6,7] => ? = 2
[1,2,5,3,6,4,7] => [7,3,5,4,6,2,1] => [7,3,6,5,4,2,1] => [1,5,2,3,4,6,7] => ? = 3
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,3,5,4,2,1] => [1,2,5,3,4,6,7] => ? = 4
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [6,7,3,5,4,2,1] => [2,1,5,3,4,6,7] => ? = 2
[1,2,5,4,6,3,7] => [7,4,5,3,6,2,1] => [7,4,6,3,5,2,1] => [1,4,2,5,3,6,7] => ? = 3
[1,2,5,4,6,7,3] => [6,4,5,3,7,2,1] => [6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => ? = 2
[1,2,5,4,7,3,6] => [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[1,2,5,4,7,6,3] => [4,5,3,6,7,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[1,2,5,6,3,4,7] => [7,5,3,6,4,2,1] => [7,5,3,6,4,2,1] => [1,3,5,2,4,6,7] => ? = 3
[1,2,5,6,3,7,4] => [5,3,6,4,7,2,1] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 2
[1,2,5,6,4,3,7] => [7,5,3,4,6,2,1] => [7,5,3,6,4,2,1] => [1,3,5,2,4,6,7] => ? = 3
[1,2,5,6,4,7,3] => [5,4,6,3,7,2,1] => [5,4,7,3,6,2,1] => [3,4,1,5,2,6,7] => ? = 2
[1,2,5,6,7,3,4] => [6,5,3,7,4,2,1] => [6,5,3,7,4,2,1] => [2,3,5,1,4,6,7] => ? = 2
[1,2,5,6,7,4,3] => [6,5,3,4,7,2,1] => [6,5,3,7,4,2,1] => [2,3,5,1,4,6,7] => ? = 2
[1,2,5,7,3,4,6] => [5,3,7,6,4,2,1] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 2
[1,2,5,7,3,6,4] => [5,3,7,4,6,2,1] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 2
[1,2,5,7,4,3,6] => [5,3,4,7,6,2,1] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 2
[1,2,5,7,4,6,3] => [5,4,7,3,6,2,1] => [5,4,7,3,6,2,1] => [3,4,1,5,2,6,7] => ? = 2
[1,2,5,7,6,3,4] => [5,3,6,7,4,2,1] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => ? = 2
Description
The number of fixed points of a permutation.
Matching statistic: St000456
(load all 5 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1 + 1
[1,2] => ([],2)
 => ([],2)
 => ? = 2 + 1
[2,1] => ([(0,1)],2)
 => ([],1)
 => ? = 0 + 1
[1,2,3] => ([],3)
 => ([],3)
 => ? = 3 + 1
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ? = 1 + 1
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ? = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ? = 0 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => ? = 4 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ? = 2 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ? = 2 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 1 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ? = 2 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? = 0 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? = 1 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ? = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 0 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ? = 5 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ? = 3 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ? = 3 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 3 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ? = 1 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? = 1 + 1
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => ? = 3 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? = 1 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000221
(load all 1286 compositions to match this statistic)
Mapping
St000221: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 58%
Values
[1] => 1
[1,2] => 2
[2,1] => 0
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 5
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 3
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 2
[1,4,2,5,3] => 1
[1,4,3,2,5] => 2
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,2,3,4,5,6,7] => ? = 7
[1,2,3,4,5,7,6] => ? = 5
[1,2,3,4,6,5,7] => ? = 5
[1,2,3,4,6,7,5] => ? = 4
[1,2,3,4,7,5,6] => ? = 4
[1,2,3,4,7,6,5] => ? = 4
[1,2,3,5,4,6,7] => ? = 5
[1,2,3,5,4,7,6] => ? = 3
[1,2,3,5,6,4,7] => ? = 4
[1,2,3,5,6,7,4] => ? = 3
[1,2,3,5,7,4,6] => ? = 3
[1,2,3,5,7,6,4] => ? = 3
[1,2,3,6,4,5,7] => ? = 4
[1,2,3,6,4,7,5] => ? = 3
[1,2,3,6,5,4,7] => ? = 4
[1,2,3,6,5,7,4] => ? = 3
[1,2,3,6,7,4,5] => ? = 3
[1,2,3,6,7,5,4] => ? = 3
[1,2,3,7,4,5,6] => ? = 3
[1,2,3,7,4,6,5] => ? = 3
[1,2,3,7,5,4,6] => ? = 3
[1,2,3,7,5,6,4] => ? = 3
[1,2,3,7,6,4,5] => ? = 3
[1,2,3,7,6,5,4] => ? = 3
[1,2,4,3,5,6,7] => ? = 5
[1,2,4,3,5,7,6] => ? = 3
[1,2,4,3,6,5,7] => ? = 3
[1,2,4,3,6,7,5] => ? = 2
[1,2,4,3,7,5,6] => ? = 2
[1,2,4,3,7,6,5] => ? = 2
[1,2,4,5,3,6,7] => ? = 4
[1,2,4,5,3,7,6] => ? = 2
[1,2,4,5,6,3,7] => ? = 3
[1,2,4,5,6,7,3] => ? = 2
[1,2,4,5,7,3,6] => ? = 2
[1,2,4,5,7,6,3] => ? = 2
[1,2,4,6,3,5,7] => ? = 3
[1,2,4,6,3,7,5] => ? = 2
[1,2,4,6,5,3,7] => ? = 3
[1,2,4,6,5,7,3] => ? = 2
[1,2,4,6,7,3,5] => ? = 2
[1,2,4,6,7,5,3] => ? = 2
[1,2,4,7,3,5,6] => ? = 2
[1,2,4,7,3,6,5] => ? = 2
[1,2,4,7,5,3,6] => ? = 2
[1,2,4,7,5,6,3] => ? = 2
[1,2,4,7,6,3,5] => ? = 2
[1,2,4,7,6,5,3] => ? = 2
[1,2,5,3,4,6,7] => ? = 4
[1,2,5,3,4,7,6] => ? = 2
Description
The number of strong fixed points of a permutation. i is called a strong fixed point of π if 1. j<i implies πj<πi, and 2. j>i implies πj>πi This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000315The number of isolated vertices of a graph. St000117The number of centered tunnels of a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000241The number of cyclical small excedances. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St000894The trace of an alternating sign matrix. St000239The number of small weak excedances. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001903The number of fixed points of a parking function. St001631The number of simple modules S with dimExt1(S,A)=1 in the incidence algebra A of the poset.