searching the database
Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000291
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
Description
The number of descents of a binary word.
Matching statistic: St000659
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 0 => 1 => 1
[1,1,0,0]
=> [1,2] => 1 => 0 => 0
[1,0,1,0,1,0]
=> [2,3,1] => 00 => 01 => 1
[1,0,1,1,0,0]
=> [2,1,3] => 01 => 10 => 1
[1,1,0,0,1,0]
=> [1,3,2] => 10 => 11 => 1
[1,1,0,1,0,0]
=> [3,1,2] => 00 => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 11 => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 000 => 001 => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 001 => 010 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 010 => 101 => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 000 => 001 => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 011 => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 100 => 011 => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 101 => 110 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 000 => 001 => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 000 => 001 => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 001 => 010 => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 110 => 111 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 100 => 011 => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 000 => 001 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0000 => 0001 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0001 => 0010 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0010 => 0101 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 0000 => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0011 => 0100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0100 => 1001 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0101 => 1010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0000 => 0001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 0000 => 0001 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 0001 => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0110 => 1011 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 0100 => 1001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 0000 => 0001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0111 => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1000 => 0011 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1001 => 0110 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 1010 => 1101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1000 => 0011 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1011 => 1100 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 0000 => 0001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 0001 => 0010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 0000 => 0001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 0000 => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0001 => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 0010 => 0101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 0000 => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0000 => 0001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0011 => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1100 => 0111 => 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,7,8,3,6] => ? => ? => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => ? => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => ? => ? => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => ? => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,6,7,8,3,5] => ? => ? => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => ? => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,8,3,5,7] => ? => ? => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,4,7,8,3,5,6] => ? => ? => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,7,8,4] => ? => ? => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [2,1,5,3,4,6,8,7] => ? => ? => ? = 3
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,1,5,3,4,6,7,8] => ? => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [2,1,3,6,4,8,5,7] => ? => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [2,1,6,7,3,4,8,5] => ? => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [2,1,6,7,3,8,4,5] => ? => ? => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,1,7,3,4,5,8,6] => ? => ? => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,5,7,2,8,6] => ? => ? => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,6,7,8,5] => ? => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,6,7,8,2,5] => ? => ? => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,4,2,5,7,6,8] => ? => ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,5,6,7,8,4] => ? => ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,5,6,7,4,8] => ? => ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,5,6,4,7,8] => ? => ? => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,7,4,8,6] => ? => ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,5,2,6,7,8,4] => ? => ? => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,5,2,6,4,7,8] => ? => ? => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,5,6,2,7,8,4] => ? => ? => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,7,8,6] => ? => ? => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,7,5,8] => ? => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,2,6,4,7,8,5] => ? => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,3,2,6,7,4,8,5] => ? => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => ? => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,2,6,8,4,5,7] => ? => ? => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,2,4,7,8,5,6] => ? => ? => ? = 2
[1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,4,5,3,8,6,7] => ? => ? => ? = 2
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,4,5,3,6,7,8] => ? => ? => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,6,3,7,8,5] => ? => ? => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,6,3,7,5,8] => ? => ? => ? = 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,6,7,3,8,5] => ? => ? => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,4,2,5,3,7,8,6] => ? => ? => ? = 2
[1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,4,5,2,6,7,3,8] => ? => ? => ? = 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,4,5,2,3,7,6,8] => ? => ? => ? = 2
[1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,4,5,7,2,3,8,6] => ? => ? => ? = 1
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,6,7,8,2,3,5] => ? => ? => ? = 1
[1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,2,8,3,5,7] => ? => ? => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> [1,2,5,3,6,4,7,8] => ? => ? => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,5,2,6,7,8,3,4] => ? => ? => ? = 1
[1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,5,6,8,2,3,4,7] => ? => ? => ? = 1
[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,2,3,4,6,8] => ? => ? => ? = 1
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,5,8,2,3,4,6,7] => ? => ? => ? = 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001011
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St001011: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001333
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => [2,1,8,4,5,6,7,3] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,7,3,8] => [2,1,7,4,5,6,3,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,6,3,8,7] => [2,1,6,4,5,3,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,6,8,3,7] => [2,1,8,4,5,6,3,7] => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,7,3,8,6] => [2,1,8,4,5,7,3,6] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,7,8,3,6] => [2,1,8,4,5,3,7,6] => ([(0,1),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => [2,1,5,4,3,6,8,7] => ([(1,4),(2,3),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,4,5,8,3,6,7] => [2,1,8,4,5,3,6,7] => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,4,5,3,6,7,8] => [2,1,5,4,3,6,7,8] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [2,1,4,3,8,6,7,5] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => [2,1,4,3,8,6,5,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,6,5,7,8] => [2,1,4,3,6,5,7,8] => ([(2,7),(3,6),(4,5)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,6,3,7,8,5] => [2,1,8,4,6,3,7,5] => ([(0,1),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => [2,1,8,4,7,6,3,5] => ([(0,1),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,6,7,8,3,5] => [2,1,8,4,3,6,7,5] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [2,1,6,4,3,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,3,8,5,7] => [2,1,8,4,6,3,5,7] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,8,3,5,7] => [2,1,8,4,3,6,5,7] => ([(0,7),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,6,3,5,7,8] => [2,1,6,4,3,5,7,8] => ([(2,3),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [2,1,4,3,5,8,7,6] => ([(1,4),(2,3),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [2,1,4,3,8,7,5,6] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,8,5,7,6] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,7,3,5,8,6] => [2,1,8,4,3,7,5,6] => ([(0,1),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,4,7,8,3,5,6] => [2,1,8,4,3,5,7,6] => ([(0,7),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,3,5,6,8,7] => [2,1,4,3,5,6,8,7] => ([(2,7),(3,6),(4,5)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,4,3,5,8,6,7] => [2,1,4,3,5,8,6,7] => ([(1,4),(2,3),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,4,3,8,5,6,7] => [2,1,4,3,8,5,6,7] => ([(0,3),(1,2),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,4,3,5,6,7,8] => [2,1,4,3,5,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,8,4] => [2,1,3,8,5,6,7,4] => ([(1,2),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,6,4,8,7] => [2,1,3,6,5,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,8,4,7] => [2,1,3,8,5,6,4,7] => ?
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,7,8,6] => [2,1,3,5,4,8,7,6] => ([(1,4),(2,3),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4,6,8,7] => [2,1,3,5,4,6,8,7] => ([(2,7),(3,6),(4,5)],8)
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [2,1,3,5,8,4,6,7] => [2,1,3,8,5,4,6,7] => ([(1,2),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,3,5,4,6,7,8] => [2,1,3,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,7,8,4] => [2,1,8,5,3,6,7,4] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,1,5,3,6,7,4,8] => [2,1,7,5,3,6,4,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [2,1,6,5,3,4,8,7] => ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,6,3,7,8,4] => [2,1,8,6,5,3,7,4] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [2,1,5,6,7,3,8,4] => [2,1,8,7,5,6,3,4] => ([(0,1),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,1,5,6,7,8,3,4] => [2,1,8,3,5,6,7,4] => ([(0,1),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,1,5,6,3,4,8,7] => [2,1,6,3,5,4,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [2,1,8,5,3,7,4,6] => ([(0,1),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [2,1,5,3,7,8,4,6] => [2,1,8,5,3,4,7,6] => ([(0,1),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [2,1,5,7,3,4,8,6] => [2,1,8,3,5,7,4,6] => ([(0,1),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [2,1,5,7,3,8,4,6] => [2,1,8,7,5,3,4,6] => ([(0,1),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [2,1,5,7,8,3,4,6] => [2,1,8,3,5,4,7,6] => ([(0,7),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [2,1,5,3,4,6,8,7] => [2,1,5,3,4,6,8,7] => ([(1,4),(2,3),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,1,5,3,8,4,6,7] => [2,1,8,5,3,4,6,7] => ([(0,1),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St000374
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [4,3,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,2,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,3,2,4,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [5,4,3,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [5,2,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [7,4,3,2,1,6,5] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [4,3,2,1,7,5,6] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [7,4,3,2,1,5,6] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => [3,2,1,6,5,4,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,3,2,1,6,5,4] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [7,3,2,1,5,4,6] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [3,2,1,4,7,6,5] => [3,2,1,4,7,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [3,2,1,7,6,4,5] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [3,2,1,7,4,6,5] => [3,2,1,7,5,6,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,2,1,6,4,5,7] => [3,2,1,6,5,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [7,3,2,1,4,6,5] => [7,4,3,2,5,6,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [7,3,2,1,6,4,5] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [3,2,1,4,7,5,6] => [3,2,1,4,7,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [3,2,1,7,4,5,6] => [3,2,1,7,5,6,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7,3,2,1,4,5,6] => [7,4,3,2,5,6,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,6,7,3,5] => [2,1,7,4,3,6,5] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,6,2,1,5,4,3] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => [7,2,1,6,5,4,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => [5,2,1,4,3,7,6] => [5,3,2,4,1,7,6] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => [7,5,2,1,4,3,6] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,1,5,4,3,6] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,1,4,3,6,7] => [5,3,2,4,1,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => [4,2,1,3,7,6,5] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,7,3,5] => [7,4,2,1,3,6,5] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,6,1,3,7,5] => [7,6,2,1,4,3,5] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,6,1,7,3,5] => [7,2,1,4,3,6,5] => [7,3,2,5,4,6,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => [7,2,1,6,4,3,5] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [4,2,1,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [7,4,2,1,3,5,6] => [7,5,4,3,2,6,1] => ? = 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [4,3,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [5,4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [5,4,3,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,2,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,3,2,4,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [5,4,3,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [5,2,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [7,4,3,2,1,6,5] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [4,3,2,1,7,5,6] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [7,4,3,2,1,5,6] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => [3,2,1,6,5,4,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,3,2,1,6,5,4] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [7,3,2,1,5,4,6] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [3,2,1,4,7,6,5] => [3,2,1,4,7,6,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [3,2,1,7,6,4,5] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [3,2,1,7,4,6,5] => [3,2,1,7,5,6,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,2,1,6,4,5,7] => [3,2,1,6,5,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [7,3,2,1,4,6,5] => [7,4,3,2,5,6,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [7,3,2,1,6,4,5] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [3,2,1,4,7,5,6] => [3,2,1,4,7,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [3,2,1,7,4,5,6] => [3,2,1,7,5,6,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7,3,2,1,4,5,6] => [7,4,3,2,5,6,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,6,7,3,5] => [2,1,7,4,3,6,5] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,6,2,1,5,4,3] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => [7,2,1,6,5,4,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => [5,2,1,4,3,7,6] => [5,3,2,4,1,7,6] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => [7,5,2,1,4,3,6] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,1,5,4,3,6] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,1,4,3,6,7] => [5,3,2,4,1,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => [4,2,1,3,7,6,5] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,7,3,5] => [7,4,2,1,3,6,5] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,6,1,3,7,5] => [7,6,2,1,4,3,5] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,6,1,7,3,5] => [7,2,1,4,3,6,5] => [7,3,2,5,4,6,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => [7,2,1,6,4,3,5] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [4,2,1,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [7,4,2,1,3,5,6] => [7,5,4,3,2,6,1] => ? = 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001665
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 80%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => {{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => {{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => {{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => {{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => {{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => {{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => {{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => {{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => {{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => {{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => {{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => {{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => {{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => {{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => {{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => {{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => {{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => {{1,2,3},{4},{5},{6},{7}}
=> [2,3,1,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => {{1,2},{3,4,5},{6,7}}
=> [2,1,4,5,3,7,6] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => {{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => {{1,2},{3,4},{5,6},{7}}
=> [2,1,4,3,6,5,7] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Matching statistic: St001737
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 80%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,4,1,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,5,1,2,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,5,1,2,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,4,5,1,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,2,4,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,5,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,4,5,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => [6,1,2,3,4,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => [5,1,2,3,4,7,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [7,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [5,7,1,2,3,4,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [4,1,2,3,6,7,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,1,2,6,7,4,5] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [7,4,5,1,2,3,6] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [6,4,5,1,2,3,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => [7,4,6,1,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [4,7,1,2,3,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [4,6,1,2,3,5,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [4,5,1,2,3,7,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [6,7,4,5,1,2,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [4,6,7,1,2,3,5] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [4,5,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [3,1,2,7,5,6,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [3,1,2,5,7,4,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,1,2,5,6,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [4,7,5,6,1,2,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [5,7,4,6,1,2,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [4,5,7,1,2,3,6] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [4,5,6,1,2,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [3,1,2,4,6,7,5] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [3,1,2,5,6,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [4,5,6,7,1,2,3] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,6,3,4,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,3,4,7,6] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [2,1,6,7,3,4,5] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,5,3,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,5,6] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,7,5,6,3,4] => ? = 2
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001060The distinguishing index of a graph. St000260The radius of a connected graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000023The number of inner peaks of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000099The number of valleys of a permutation, including the boundary. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000862The number of parts of the shifted shape of a permutation. St000007The number of saliances of the permutation. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001597The Frobenius rank of a skew partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001741The largest integer such that all patterns of this size are contained in the permutation. St000542The number of left-to-right-minima of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!