Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000340
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => [1,1,0,0]
 => 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 1
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 1
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000691
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => 10 => 1
[-1] => 1 => [1,1] => 11 => 0
[1,2] => 00 => [3] => 100 => 1
[1,-2] => 01 => [2,1] => 101 => 2
[-1,2] => 10 => [1,2] => 110 => 1
[-1,-2] => 11 => [1,1,1] => 111 => 0
[2,1] => 00 => [3] => 100 => 1
[2,-1] => 01 => [2,1] => 101 => 2
[-2,1] => 10 => [1,2] => 110 => 1
[-2,-1] => 11 => [1,1,1] => 111 => 0
[1,2,3] => 000 => [4] => 1000 => 1
[1,2,-3] => 001 => [3,1] => 1001 => 2
[1,-2,3] => 010 => [2,2] => 1010 => 3
[1,-2,-3] => 011 => [2,1,1] => 1011 => 2
[-1,2,3] => 100 => [1,3] => 1100 => 1
[-1,2,-3] => 101 => [1,2,1] => 1101 => 2
[-1,-2,3] => 110 => [1,1,2] => 1110 => 1
[-1,-2,-3] => 111 => [1,1,1,1] => 1111 => 0
[1,3,2] => 000 => [4] => 1000 => 1
[1,3,-2] => 001 => [3,1] => 1001 => 2
[1,-3,2] => 010 => [2,2] => 1010 => 3
[1,-3,-2] => 011 => [2,1,1] => 1011 => 2
[-1,3,2] => 100 => [1,3] => 1100 => 1
[-1,3,-2] => 101 => [1,2,1] => 1101 => 2
[-1,-3,2] => 110 => [1,1,2] => 1110 => 1
[-1,-3,-2] => 111 => [1,1,1,1] => 1111 => 0
[2,1,3] => 000 => [4] => 1000 => 1
[2,1,-3] => 001 => [3,1] => 1001 => 2
[2,-1,3] => 010 => [2,2] => 1010 => 3
[2,-1,-3] => 011 => [2,1,1] => 1011 => 2
[-2,1,3] => 100 => [1,3] => 1100 => 1
[-2,1,-3] => 101 => [1,2,1] => 1101 => 2
[-2,-1,3] => 110 => [1,1,2] => 1110 => 1
[-2,-1,-3] => 111 => [1,1,1,1] => 1111 => 0
[2,3,1] => 000 => [4] => 1000 => 1
[2,3,-1] => 001 => [3,1] => 1001 => 2
[2,-3,1] => 010 => [2,2] => 1010 => 3
[2,-3,-1] => 011 => [2,1,1] => 1011 => 2
[-2,3,1] => 100 => [1,3] => 1100 => 1
[-2,3,-1] => 101 => [1,2,1] => 1101 => 2
[-2,-3,1] => 110 => [1,1,2] => 1110 => 1
[-2,-3,-1] => 111 => [1,1,1,1] => 1111 => 0
[3,1,2] => 000 => [4] => 1000 => 1
[3,1,-2] => 001 => [3,1] => 1001 => 2
[3,-1,2] => 010 => [2,2] => 1010 => 3
[3,-1,-2] => 011 => [2,1,1] => 1011 => 2
[-3,1,2] => 100 => [1,3] => 1100 => 1
[-3,1,-2] => 101 => [1,2,1] => 1101 => 2
[-3,-1,2] => 110 => [1,1,2] => 1110 => 1
[-3,-1,-2] => 111 => [1,1,1,1] => 1111 => 0
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001870
(load all 2 compositions to match this statistic)
Mapping
Mp00161: Signed permutations —reverse⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [-1] => 1
[-1] => [-1] => [1] => 0
[1,2] => [2,1] => [-2,-1] => 1
[1,-2] => [-2,1] => [2,-1] => 2
[-1,2] => [2,-1] => [-2,1] => 1
[-1,-2] => [-2,-1] => [2,1] => 0
[2,1] => [1,2] => [-1,-2] => 1
[2,-1] => [-1,2] => [1,-2] => 2
[-2,1] => [1,-2] => [-1,2] => 1
[-2,-1] => [-1,-2] => [1,2] => 0
[1,2,3] => [3,2,1] => [-3,-2,-1] => 1
[1,2,-3] => [-3,2,1] => [3,-2,-1] => 2
[1,-2,3] => [3,-2,1] => [-3,2,-1] => 3
[1,-2,-3] => [-3,-2,1] => [3,2,-1] => 2
[-1,2,3] => [3,2,-1] => [-3,-2,1] => 1
[-1,2,-3] => [-3,2,-1] => [3,-2,1] => 2
[-1,-2,3] => [3,-2,-1] => [-3,2,1] => 1
[-1,-2,-3] => [-3,-2,-1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [-2,-3,-1] => 1
[1,3,-2] => [-2,3,1] => [2,-3,-1] => 2
[1,-3,2] => [2,-3,1] => [-2,3,-1] => 3
[1,-3,-2] => [-2,-3,1] => [2,3,-1] => 2
[-1,3,2] => [2,3,-1] => [-2,-3,1] => 1
[-1,3,-2] => [-2,3,-1] => [2,-3,1] => 2
[-1,-3,2] => [2,-3,-1] => [-2,3,1] => 1
[-1,-3,-2] => [-2,-3,-1] => [2,3,1] => 0
[2,1,3] => [3,1,2] => [-3,-1,-2] => 1
[2,1,-3] => [-3,1,2] => [3,-1,-2] => 2
[2,-1,3] => [3,-1,2] => [-3,1,-2] => 3
[2,-1,-3] => [-3,-1,2] => [3,1,-2] => 2
[-2,1,3] => [3,1,-2] => [-3,-1,2] => 1
[-2,1,-3] => [-3,1,-2] => [3,-1,2] => 2
[-2,-1,3] => [3,-1,-2] => [-3,1,2] => 1
[-2,-1,-3] => [-3,-1,-2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [-1,-3,-2] => 1
[2,3,-1] => [-1,3,2] => [1,-3,-2] => 2
[2,-3,1] => [1,-3,2] => [-1,3,-2] => 3
[2,-3,-1] => [-1,-3,2] => [1,3,-2] => 2
[-2,3,1] => [1,3,-2] => [-1,-3,2] => 1
[-2,3,-1] => [-1,3,-2] => [1,-3,2] => 2
[-2,-3,1] => [1,-3,-2] => [-1,3,2] => 1
[-2,-3,-1] => [-1,-3,-2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [-2,-1,-3] => 1
[3,1,-2] => [-2,1,3] => [2,-1,-3] => 2
[3,-1,2] => [2,-1,3] => [-2,1,-3] => 3
[3,-1,-2] => [-2,-1,3] => [2,1,-3] => 2
[-3,1,2] => [2,1,-3] => [-2,-1,3] => 1
[-3,1,-2] => [-2,1,-3] => [2,-1,3] => 2
[-3,-1,2] => [2,-1,-3] => [-2,1,3] => 1
[-3,-1,-2] => [-2,-1,-3] => [2,1,3] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 1
[1,2,3,4,-5] => [-5,4,3,2,1] => [5,-4,-3,-2,-1] => ? = 2
[1,2,3,-4,5] => [5,-4,3,2,1] => [-5,4,-3,-2,-1] => ? = 3
[1,2,3,-4,-5] => [-5,-4,3,2,1] => [5,4,-3,-2,-1] => ? = 2
[1,2,-3,4,5] => [5,4,-3,2,1] => [-5,-4,3,-2,-1] => ? = 3
[1,2,-3,4,-5] => [-5,4,-3,2,1] => [5,-4,3,-2,-1] => ? = 4
[1,2,-3,-4,5] => [5,-4,-3,2,1] => [-5,4,3,-2,-1] => ? = 3
[1,2,-3,-4,-5] => [-5,-4,-3,2,1] => [5,4,3,-2,-1] => ? = 2
[1,-2,3,4,5] => [5,4,3,-2,1] => [-5,-4,-3,2,-1] => ? = 3
[1,-2,3,4,-5] => [-5,4,3,-2,1] => [5,-4,-3,2,-1] => ? = 4
[1,-2,3,-4,5] => [5,-4,3,-2,1] => [-5,4,-3,2,-1] => ? = 5
[1,-2,3,-4,-5] => [-5,-4,3,-2,1] => [5,4,-3,2,-1] => ? = 4
[1,-2,-3,4,5] => [5,4,-3,-2,1] => [-5,-4,3,2,-1] => ? = 3
[1,-2,-3,4,-5] => [-5,4,-3,-2,1] => [5,-4,3,2,-1] => ? = 4
[1,-2,-3,-4,5] => [5,-4,-3,-2,1] => [-5,4,3,2,-1] => ? = 3
[1,-2,-3,-4,-5] => [-5,-4,-3,-2,1] => [5,4,3,2,-1] => ? = 2
[-1,2,3,4,5] => [5,4,3,2,-1] => [-5,-4,-3,-2,1] => ? = 1
[-1,2,3,4,-5] => [-5,4,3,2,-1] => [5,-4,-3,-2,1] => ? = 2
[-1,2,3,-4,5] => [5,-4,3,2,-1] => [-5,4,-3,-2,1] => ? = 3
[-1,2,3,-4,-5] => [-5,-4,3,2,-1] => [5,4,-3,-2,1] => ? = 2
[-1,2,-3,4,5] => [5,4,-3,2,-1] => [-5,-4,3,-2,1] => ? = 3
[-1,2,-3,4,-5] => [-5,4,-3,2,-1] => [5,-4,3,-2,1] => ? = 4
[-1,2,-3,-4,5] => [5,-4,-3,2,-1] => [-5,4,3,-2,1] => ? = 3
[-1,2,-3,-4,-5] => [-5,-4,-3,2,-1] => [5,4,3,-2,1] => ? = 2
[-1,-2,3,4,5] => [5,4,3,-2,-1] => [-5,-4,-3,2,1] => ? = 1
[-1,-2,3,4,-5] => [-5,4,3,-2,-1] => [5,-4,-3,2,1] => ? = 2
[-1,-2,3,-4,5] => [5,-4,3,-2,-1] => [-5,4,-3,2,1] => ? = 3
[-1,-2,3,-4,-5] => [-5,-4,3,-2,-1] => [5,4,-3,2,1] => ? = 2
[-1,-2,-3,4,5] => [5,4,-3,-2,-1] => [-5,-4,3,2,1] => ? = 1
[-1,-2,-3,4,-5] => [-5,4,-3,-2,-1] => [5,-4,3,2,1] => ? = 2
[-1,-2,-3,-4,5] => [5,-4,-3,-2,-1] => [-5,4,3,2,1] => ? = 1
[-1,-2,-3,-4,-5] => [-5,-4,-3,-2,-1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [4,5,3,2,1] => [-4,-5,-3,-2,-1] => ? = 1
[1,2,3,5,-4] => [-4,5,3,2,1] => [4,-5,-3,-2,-1] => ? = 2
[1,2,3,-5,4] => [4,-5,3,2,1] => [-4,5,-3,-2,-1] => ? = 3
[1,2,3,-5,-4] => [-4,-5,3,2,1] => [4,5,-3,-2,-1] => ? = 2
[1,2,-3,5,4] => [4,5,-3,2,1] => [-4,-5,3,-2,-1] => ? = 3
[1,2,-3,5,-4] => [-4,5,-3,2,1] => [4,-5,3,-2,-1] => ? = 4
[1,2,-3,-5,4] => [4,-5,-3,2,1] => [-4,5,3,-2,-1] => ? = 3
[1,2,-3,-5,-4] => [-4,-5,-3,2,1] => [4,5,3,-2,-1] => ? = 2
[1,-2,3,5,4] => [4,5,3,-2,1] => [-4,-5,-3,2,-1] => ? = 3
[1,-2,3,5,-4] => [-4,5,3,-2,1] => [4,-5,-3,2,-1] => ? = 4
[1,-2,3,-5,4] => [4,-5,3,-2,1] => [-4,5,-3,2,-1] => ? = 5
[1,-2,3,-5,-4] => [-4,-5,3,-2,1] => [4,5,-3,2,-1] => ? = 4
[1,-2,-3,5,4] => [4,5,-3,-2,1] => [-4,-5,3,2,-1] => ? = 3
[1,-2,-3,5,-4] => [-4,5,-3,-2,1] => [4,-5,3,2,-1] => ? = 4
[1,-2,-3,-5,4] => [4,-5,-3,-2,1] => [-4,5,3,2,-1] => ? = 3
[1,-2,-3,-5,-4] => [-4,-5,-3,-2,1] => [4,5,3,2,-1] => ? = 2
[-1,2,3,5,4] => [4,5,3,2,-1] => [-4,-5,-3,-2,1] => ? = 1
[-1,2,3,5,-4] => [-4,5,3,2,-1] => [4,-5,-3,-2,1] => ? = 2
Description
The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.