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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000355
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[3,1,1] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,2,1] => [3,1,2] => [1,3,2] => [2,3,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [3,1,2] => [1,3,2] => [2,3,1] => 0
[3,1,2] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,1] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 0
[3,1,1,1] => [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => [4,1,3,2] => 0
[1,4,1,1] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[4,1,1,1] => [2,3,4,1] => [2,3,4,1] => [2,1,4,3] => 2
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,2,1,1] => [3,1,2,4] => [1,3,2,4] => [1,2,4,3] => 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,1,2] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 0
[1,3,2,1] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 0
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[2,3,1,1] => [3,1,2,4] => [1,3,2,4] => [1,2,4,3] => 0
[3,1,1,2] => [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 0
[3,1,2,1] => [2,3,1,4] => [2,3,1,4] => [1,4,2,3] => 0
Description
The number of occurrences of the pattern 21-3.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St001857
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,2,1] => [3,1,2] => [2,3,1] => [2,3,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [2,3,1] => [2,3,1] => 0
[3,1,2] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [3,2,1] => [3,1,2] => [3,1,2] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ? = 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,3,1,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ? = 0
[3,1,1,1] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ? = 0
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ? = 0
[1,4,1,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ? = 0
[4,1,1,1] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => ? = 2
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ? = 1
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ? = 1
[2,2,1,1] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => ? = 0
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,3,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ? = 0
[1,3,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ? = 0
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ? = 1
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ? = 1
[2,3,1,1] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => ? = 0
[3,1,1,2] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ? = 0
[3,1,2,1] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ? = 0
[3,2,1,1] => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => ? = 0
[1,1,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,1,4,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ? = 0
[1,2,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ? = 0
[1,4,1,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ? = 0
[1,4,2,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ? = 0
[2,1,1,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ? = 1
[2,1,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ? = 0
[2,4,1,1] => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => ? = 1
[4,1,1,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => ? = 2
[4,1,2,1] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => ? = 2
[4,2,1,1] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => ? = 2
[1,1,3,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
[1,3,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ? = 0
[1,3,3,1] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => ? = 0
[3,1,1,3] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ? = 0
[3,1,3,1] => [2,4,1,3] => [3,2,4,1] => [3,2,4,1] => ? = 0
[3,3,1,1] => [3,4,1,2] => [2,4,3,1] => [2,4,3,1] => ? = 1
[1,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St001722
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1,0]
=> 10 => 1 = 0 + 1
[1,1] => [1,2] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> 1100 => 1 = 0 + 1
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,1,1] => [2,3,1] => [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[2,2,1] => [3,1,2] => [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 2 + 1
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 0 + 1
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 0 + 1
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 0 + 1
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[2,4,1,1] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 1 + 1
[4,1,1,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 2 + 1
[4,1,2,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 2 + 1
[4,2,1,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 2 + 1
[1,1,3,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,3,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[1,3,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[3,1,1,3] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 0 + 1
[3,1,3,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 11011000 => ? = 0 + 1
[3,3,1,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 11101000 => ? = 1 + 1
[1,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
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