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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
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(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => 1 = 2 - 1
[1,2] => 1 => [1,1] => 1 = 2 - 1
[2,1] => 0 => [2] => 2 = 3 - 1
[1,2,3] => 11 => [1,1,1] => 1 = 2 - 1
[1,3,2] => 10 => [1,2] => 1 = 2 - 1
[2,1,3] => 01 => [2,1] => 2 = 3 - 1
[2,3,1] => 00 => [3] => 3 = 4 - 1
[3,1,2] => 00 => [3] => 3 = 4 - 1
[3,2,1] => 00 => [3] => 3 = 4 - 1
[1,2,3,4] => 111 => [1,1,1,1] => 1 = 2 - 1
[1,2,4,3] => 110 => [1,1,2] => 1 = 2 - 1
[1,3,2,4] => 101 => [1,2,1] => 1 = 2 - 1
[1,3,4,2] => 100 => [1,3] => 1 = 2 - 1
[1,4,2,3] => 100 => [1,3] => 1 = 2 - 1
[1,4,3,2] => 100 => [1,3] => 1 = 2 - 1
[2,1,3,4] => 011 => [2,1,1] => 2 = 3 - 1
[2,1,4,3] => 010 => [2,2] => 2 = 3 - 1
[2,3,1,4] => 001 => [3,1] => 3 = 4 - 1
[2,3,4,1] => 000 => [4] => 4 = 5 - 1
[2,4,1,3] => 000 => [4] => 4 = 5 - 1
[2,4,3,1] => 000 => [4] => 4 = 5 - 1
[3,1,2,4] => 001 => [3,1] => 3 = 4 - 1
[3,1,4,2] => 000 => [4] => 4 = 5 - 1
[3,2,1,4] => 001 => [3,1] => 3 = 4 - 1
[3,2,4,1] => 000 => [4] => 4 = 5 - 1
[3,4,1,2] => 000 => [4] => 4 = 5 - 1
[3,4,2,1] => 000 => [4] => 4 = 5 - 1
[4,1,2,3] => 000 => [4] => 4 = 5 - 1
[4,1,3,2] => 000 => [4] => 4 = 5 - 1
[4,2,1,3] => 000 => [4] => 4 = 5 - 1
[4,2,3,1] => 000 => [4] => 4 = 5 - 1
[4,3,1,2] => 000 => [4] => 4 = 5 - 1
[4,3,2,1] => 000 => [4] => 4 = 5 - 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => 1 = 2 - 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => 1 = 2 - 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => 1 = 2 - 1
[1,2,4,5,3] => 1100 => [1,1,3] => 1 = 2 - 1
[1,2,5,3,4] => 1100 => [1,1,3] => 1 = 2 - 1
[1,2,5,4,3] => 1100 => [1,1,3] => 1 = 2 - 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => 1 = 2 - 1
[1,3,2,5,4] => 1010 => [1,2,2] => 1 = 2 - 1
[1,3,4,2,5] => 1001 => [1,3,1] => 1 = 2 - 1
[1,3,4,5,2] => 1000 => [1,4] => 1 = 2 - 1
[1,3,5,2,4] => 1000 => [1,4] => 1 = 2 - 1
[1,3,5,4,2] => 1000 => [1,4] => 1 = 2 - 1
[1,4,2,3,5] => 1001 => [1,3,1] => 1 = 2 - 1
[1,4,2,5,3] => 1000 => [1,4] => 1 = 2 - 1
[1,4,3,2,5] => 1001 => [1,3,1] => 1 = 2 - 1
[1,4,3,5,2] => 1000 => [1,4] => 1 = 2 - 1
[1,4,5,2,3] => 1000 => [1,4] => 1 = 2 - 1
Description
The first part of an integer composition.
Matching statistic: St000439
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 2
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 2
[2,1] => 0 => [2] => [1,1,0,0]
=> 3
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 4
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 4
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 4
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000326
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 2 - 1
[1,2] => 1 => 1 = 2 - 1
[2,1] => 0 => 2 = 3 - 1
[1,2,3] => 11 => 1 = 2 - 1
[1,3,2] => 10 => 1 = 2 - 1
[2,1,3] => 01 => 2 = 3 - 1
[2,3,1] => 00 => 3 = 4 - 1
[3,1,2] => 00 => 3 = 4 - 1
[3,2,1] => 00 => 3 = 4 - 1
[1,2,3,4] => 111 => 1 = 2 - 1
[1,2,4,3] => 110 => 1 = 2 - 1
[1,3,2,4] => 101 => 1 = 2 - 1
[1,3,4,2] => 100 => 1 = 2 - 1
[1,4,2,3] => 100 => 1 = 2 - 1
[1,4,3,2] => 100 => 1 = 2 - 1
[2,1,3,4] => 011 => 2 = 3 - 1
[2,1,4,3] => 010 => 2 = 3 - 1
[2,3,1,4] => 001 => 3 = 4 - 1
[2,3,4,1] => 000 => 4 = 5 - 1
[2,4,1,3] => 000 => 4 = 5 - 1
[2,4,3,1] => 000 => 4 = 5 - 1
[3,1,2,4] => 001 => 3 = 4 - 1
[3,1,4,2] => 000 => 4 = 5 - 1
[3,2,1,4] => 001 => 3 = 4 - 1
[3,2,4,1] => 000 => 4 = 5 - 1
[3,4,1,2] => 000 => 4 = 5 - 1
[3,4,2,1] => 000 => 4 = 5 - 1
[4,1,2,3] => 000 => 4 = 5 - 1
[4,1,3,2] => 000 => 4 = 5 - 1
[4,2,1,3] => 000 => 4 = 5 - 1
[4,2,3,1] => 000 => 4 = 5 - 1
[4,3,1,2] => 000 => 4 = 5 - 1
[4,3,2,1] => 000 => 4 = 5 - 1
[1,2,3,4,5] => 1111 => 1 = 2 - 1
[1,2,3,5,4] => 1110 => 1 = 2 - 1
[1,2,4,3,5] => 1101 => 1 = 2 - 1
[1,2,4,5,3] => 1100 => 1 = 2 - 1
[1,2,5,3,4] => 1100 => 1 = 2 - 1
[1,2,5,4,3] => 1100 => 1 = 2 - 1
[1,3,2,4,5] => 1011 => 1 = 2 - 1
[1,3,2,5,4] => 1010 => 1 = 2 - 1
[1,3,4,2,5] => 1001 => 1 = 2 - 1
[1,3,4,5,2] => 1000 => 1 = 2 - 1
[1,3,5,2,4] => 1000 => 1 = 2 - 1
[1,3,5,4,2] => 1000 => 1 = 2 - 1
[1,4,2,3,5] => 1001 => 1 = 2 - 1
[1,4,2,5,3] => 1000 => 1 = 2 - 1
[1,4,3,2,5] => 1001 => 1 = 2 - 1
[1,4,3,5,2] => 1000 => 1 = 2 - 1
[1,4,5,2,3] => 1000 => 1 = 2 - 1
[1,4,5,3,2] => 1000 => 1 = 2 - 1
[] => => ? = 2 - 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? = 2 - 2
[1,2] => 1 => 0 => 0 = 2 - 2
[2,1] => 0 => 1 => 1 = 3 - 2
[1,2,3] => 11 => 00 => 0 = 2 - 2
[1,3,2] => 10 => 01 => 0 = 2 - 2
[2,1,3] => 01 => 10 => 1 = 3 - 2
[2,3,1] => 00 => 11 => 2 = 4 - 2
[3,1,2] => 00 => 11 => 2 = 4 - 2
[3,2,1] => 00 => 11 => 2 = 4 - 2
[1,2,3,4] => 111 => 000 => 0 = 2 - 2
[1,2,4,3] => 110 => 001 => 0 = 2 - 2
[1,3,2,4] => 101 => 010 => 0 = 2 - 2
[1,3,4,2] => 100 => 011 => 0 = 2 - 2
[1,4,2,3] => 100 => 011 => 0 = 2 - 2
[1,4,3,2] => 100 => 011 => 0 = 2 - 2
[2,1,3,4] => 011 => 100 => 1 = 3 - 2
[2,1,4,3] => 010 => 101 => 1 = 3 - 2
[2,3,1,4] => 001 => 110 => 2 = 4 - 2
[2,3,4,1] => 000 => 111 => 3 = 5 - 2
[2,4,1,3] => 000 => 111 => 3 = 5 - 2
[2,4,3,1] => 000 => 111 => 3 = 5 - 2
[3,1,2,4] => 001 => 110 => 2 = 4 - 2
[3,1,4,2] => 000 => 111 => 3 = 5 - 2
[3,2,1,4] => 001 => 110 => 2 = 4 - 2
[3,2,4,1] => 000 => 111 => 3 = 5 - 2
[3,4,1,2] => 000 => 111 => 3 = 5 - 2
[3,4,2,1] => 000 => 111 => 3 = 5 - 2
[4,1,2,3] => 000 => 111 => 3 = 5 - 2
[4,1,3,2] => 000 => 111 => 3 = 5 - 2
[4,2,1,3] => 000 => 111 => 3 = 5 - 2
[4,2,3,1] => 000 => 111 => 3 = 5 - 2
[4,3,1,2] => 000 => 111 => 3 = 5 - 2
[4,3,2,1] => 000 => 111 => 3 = 5 - 2
[1,2,3,4,5] => 1111 => 0000 => 0 = 2 - 2
[1,2,3,5,4] => 1110 => 0001 => 0 = 2 - 2
[1,2,4,3,5] => 1101 => 0010 => 0 = 2 - 2
[1,2,4,5,3] => 1100 => 0011 => 0 = 2 - 2
[1,2,5,3,4] => 1100 => 0011 => 0 = 2 - 2
[1,2,5,4,3] => 1100 => 0011 => 0 = 2 - 2
[1,3,2,4,5] => 1011 => 0100 => 0 = 2 - 2
[1,3,2,5,4] => 1010 => 0101 => 0 = 2 - 2
[1,3,4,2,5] => 1001 => 0110 => 0 = 2 - 2
[1,3,4,5,2] => 1000 => 0111 => 0 = 2 - 2
[1,3,5,2,4] => 1000 => 0111 => 0 = 2 - 2
[1,3,5,4,2] => 1000 => 0111 => 0 = 2 - 2
[1,4,2,3,5] => 1001 => 0110 => 0 = 2 - 2
[1,4,2,5,3] => 1000 => 0111 => 0 = 2 - 2
[1,4,3,2,5] => 1001 => 0110 => 0 = 2 - 2
[1,4,3,5,2] => 1000 => 0111 => 0 = 2 - 2
[1,4,5,2,3] => 1000 => 0111 => 0 = 2 - 2
[1,4,5,3,2] => 1000 => 0111 => 0 = 2 - 2
[] => => => ? = 2 - 2
Description
The number of leading ones in a binary word.
Matching statistic: St000054
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[2,1] => [2,1] => [2,1] => [2,1] => 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 4 - 1
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 4 - 1
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 4 - 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 4 - 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1 = 2 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 2 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 2 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 2 - 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2 - 1
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,5,7,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2 - 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2 - 1
[1,2,3,6,5,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,7,4,5,6] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,7,4,6,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,7,5,4,6] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,7,6,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 2 - 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2 - 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 2 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 2 - 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 2 - 1
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 2 - 1
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 2 - 1
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 2 - 1
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,5,7,3,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 2 - 1
[1,2,4,6,3,7,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 2 - 1
[1,2,4,6,5,7,3] => [1,2,7,5,4,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,6,7,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,7,3,5,6] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,7,3,6,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,7,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,7,6,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,4,7,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 2 - 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 2 - 1
[1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 2 - 1
[1,2,5,3,6,7,4] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
[1,2,5,3,7,4,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 2 - 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000505
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 1 = 2 - 1
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1 = 2 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 2 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4 = 5 - 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 2 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[4,3,2,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[3,4,2,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[4,2,3,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[7,2,1,3,4,5,6,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 8 - 1
[3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 4 - 1
[2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 4 - 1
[3,1,2,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 4 - 1
[2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 3 - 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 2 - 1
[4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 9 - 1
[3,4,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 9 - 1
[2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 9 - 1
[4,3,2,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ?
=> ? = 9 - 1
[3,4,2,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ?
=> ? = 9 - 1
[3,4,2,6,7,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ?
=> ? = 9 - 1
[2,4,3,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ?
=> ? = 9 - 1
[2,4,3,6,7,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ?
=> ? = 9 - 1
[3,2,4,5,7,6,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ?
=> ? = 9 - 1
[2,3,4,5,7,6,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ?
=> ? = 9 - 1
[4,3,2,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ?
=> ? = 9 - 1
[2,4,3,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ?
=> ? = 9 - 1
[3,2,5,4,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ?
=> ? = 9 - 1
[2,3,5,4,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ?
=> ? = 9 - 1
[1,3,5,4,8,7,6,2] => [1,8,4,3,7,6,5,2] => ?
=> ?
=> ? = 2 - 1
[1,3,4,5,6,8,7,2] => [1,8,3,4,5,7,6,2] => ?
=> ?
=> ? = 2 - 1
[1,2,5,4,6,8,7,3] => [1,2,8,4,5,7,6,3] => ?
=> ?
=> ? = 2 - 1
[1,2,4,5,6,8,7,3] => [1,2,8,4,5,7,6,3] => ?
=> ?
=> ? = 2 - 1
[1,2,5,4,7,6,8,3] => [1,2,8,4,6,5,7,3] => ?
=> ?
=> ? = 2 - 1
[1,3,2,8,7,6,5,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,6,8,7,5,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,6,7,8,5,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,5,8,7,6,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,5,7,8,6,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,6,5,8,7,4] => [1,3,2,8,5,7,6,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,5,6,8,7,4] => [1,3,2,8,5,7,6,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,3,2,7,6,5,8,4] => [1,3,2,8,6,5,7,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 - 1
[2,1,3,8,7,6,5,4] => [2,1,3,8,7,6,5,4] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 3 - 1
[2,1,3,7,8,6,5,4] => [2,1,3,8,7,6,5,4] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 3 - 1
[2,1,3,6,8,7,5,4] => [2,1,3,8,7,6,5,4] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 3 - 1
[2,1,3,6,5,8,7,4] => [2,1,3,8,5,7,6,4] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 3 - 1
[2,1,3,5,6,7,8,4] => [2,1,3,8,5,6,7,4] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 3 - 1
[1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,2,3,7,8,6,5,4] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
[1,2,3,6,8,7,5,4] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 2 - 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1] => => => [1] => 1 = 2 - 1
[1,2] => 1 => 1 => [1,1] => 1 = 2 - 1
[2,1] => 0 => 0 => [2] => 2 = 3 - 1
[1,2,3] => 11 => 11 => [1,1,1] => 1 = 2 - 1
[1,3,2] => 10 => 01 => [2,1] => 1 = 2 - 1
[2,1,3] => 01 => 10 => [1,2] => 2 = 3 - 1
[2,3,1] => 00 => 00 => [3] => 3 = 4 - 1
[3,1,2] => 00 => 00 => [3] => 3 = 4 - 1
[3,2,1] => 00 => 00 => [3] => 3 = 4 - 1
[1,2,3,4] => 111 => 111 => [1,1,1,1] => 1 = 2 - 1
[1,2,4,3] => 110 => 011 => [2,1,1] => 1 = 2 - 1
[1,3,2,4] => 101 => 101 => [1,2,1] => 1 = 2 - 1
[1,3,4,2] => 100 => 001 => [3,1] => 1 = 2 - 1
[1,4,2,3] => 100 => 001 => [3,1] => 1 = 2 - 1
[1,4,3,2] => 100 => 001 => [3,1] => 1 = 2 - 1
[2,1,3,4] => 011 => 110 => [1,1,2] => 2 = 3 - 1
[2,1,4,3] => 010 => 010 => [2,2] => 2 = 3 - 1
[2,3,1,4] => 001 => 100 => [1,3] => 3 = 4 - 1
[2,3,4,1] => 000 => 000 => [4] => 4 = 5 - 1
[2,4,1,3] => 000 => 000 => [4] => 4 = 5 - 1
[2,4,3,1] => 000 => 000 => [4] => 4 = 5 - 1
[3,1,2,4] => 001 => 100 => [1,3] => 3 = 4 - 1
[3,1,4,2] => 000 => 000 => [4] => 4 = 5 - 1
[3,2,1,4] => 001 => 100 => [1,3] => 3 = 4 - 1
[3,2,4,1] => 000 => 000 => [4] => 4 = 5 - 1
[3,4,1,2] => 000 => 000 => [4] => 4 = 5 - 1
[3,4,2,1] => 000 => 000 => [4] => 4 = 5 - 1
[4,1,2,3] => 000 => 000 => [4] => 4 = 5 - 1
[4,1,3,2] => 000 => 000 => [4] => 4 = 5 - 1
[4,2,1,3] => 000 => 000 => [4] => 4 = 5 - 1
[4,2,3,1] => 000 => 000 => [4] => 4 = 5 - 1
[4,3,1,2] => 000 => 000 => [4] => 4 = 5 - 1
[4,3,2,1] => 000 => 000 => [4] => 4 = 5 - 1
[1,2,3,4,5] => 1111 => 1111 => [1,1,1,1,1] => 1 = 2 - 1
[1,2,3,5,4] => 1110 => 0111 => [2,1,1,1] => 1 = 2 - 1
[1,2,4,3,5] => 1101 => 1011 => [1,2,1,1] => 1 = 2 - 1
[1,2,4,5,3] => 1100 => 0011 => [3,1,1] => 1 = 2 - 1
[1,2,5,3,4] => 1100 => 0011 => [3,1,1] => 1 = 2 - 1
[1,2,5,4,3] => 1100 => 0011 => [3,1,1] => 1 = 2 - 1
[1,3,2,4,5] => 1011 => 1101 => [1,1,2,1] => 1 = 2 - 1
[1,3,2,5,4] => 1010 => 0101 => [2,2,1] => 1 = 2 - 1
[1,3,4,2,5] => 1001 => 1001 => [1,3,1] => 1 = 2 - 1
[1,3,4,5,2] => 1000 => 0001 => [4,1] => 1 = 2 - 1
[1,3,5,2,4] => 1000 => 0001 => [4,1] => 1 = 2 - 1
[1,3,5,4,2] => 1000 => 0001 => [4,1] => 1 = 2 - 1
[1,4,2,3,5] => 1001 => 1001 => [1,3,1] => 1 = 2 - 1
[1,4,2,5,3] => 1000 => 0001 => [4,1] => 1 = 2 - 1
[1,4,3,2,5] => 1001 => 1001 => [1,3,1] => 1 = 2 - 1
[1,4,3,5,2] => 1000 => 0001 => [4,1] => 1 = 2 - 1
[1,4,5,2,3] => 1000 => 0001 => [4,1] => 1 = 2 - 1
[8,7,6,5,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,8,6,5,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,8,5,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,7,8,5,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,5,6,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,8,5,6,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,6,5,7,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,5,6,7,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,5,8,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,7,5,8,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,5,6,8,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,5,7,8,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[5,6,7,8,4,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,6,4,5,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,8,6,4,5,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,6,7,4,5,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,5,4,6,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,4,5,6,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,5,4,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,7,5,4,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,5,6,4,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,5,7,4,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[5,6,7,4,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,4,5,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,7,4,5,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,5,4,6,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,4,5,6,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,5,4,7,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[5,6,4,7,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,4,5,7,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[5,4,6,7,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[4,5,6,7,8,3,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,6,5,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,8,6,5,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,6,7,5,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,8,5,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,5,6,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,8,5,6,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,5,6,7,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,5,8,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,7,5,8,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,5,7,8,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[5,6,7,8,3,4,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,8,4,3,5,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,6,3,4,5,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,8,6,3,4,5,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,6,7,3,4,5,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[7,6,8,3,4,5,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[6,7,8,3,4,5,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
[8,7,5,4,3,6,2,1] => 0000000 => 0000000 => [8] => ? = 9 - 1
Description
The last part of an integer composition.
Matching statistic: St000025
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 70%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 70%
Values
[1] => => [1] => [1,0]
=> 1 = 2 - 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2 = 3 - 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[8,7,6,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,8,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,6,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,5,6,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,5,7,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,5,6,7,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,5,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,5,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,5,6,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,7,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,7,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,6,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,7,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,4,6,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,4,5,6,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,5,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,5,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,5,6,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,7,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,7,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,4,5,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,4,5,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,5,4,6,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,4,5,6,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,4,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,4,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,4,5,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,4,6,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[4,5,6,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,6,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,7,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,6,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,5,6,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,5,6,7,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,5,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,5,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,7,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,7,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,4,3,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,6,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,7,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,8,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,4,3,6,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 109 compositions to match this statistic)
(load all 109 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 70%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 70%
Values
[1] => => [1] => [1,0]
=> 1 = 2 - 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2 = 3 - 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[8,7,6,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,8,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,6,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,5,6,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,5,7,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,5,6,7,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,5,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,5,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,5,6,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,7,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,7,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,6,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,7,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,4,6,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,4,5,6,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,5,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,5,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,5,6,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,7,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,7,4,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,4,5,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,4,5,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,5,4,6,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,4,5,6,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,4,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,4,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,4,5,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,4,6,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[4,5,6,7,8,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,6,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,7,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,6,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,5,6,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,5,6,7,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,5,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,5,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,5,7,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[5,6,7,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,4,3,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,6,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,8,6,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,6,7,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[7,6,8,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[6,7,8,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[8,7,5,4,3,6,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000066
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [1,2] => [[1,0],[0,1]]
=> 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [2,1] => [[0,1],[1,0]]
=> 2 = 3 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 2 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3 = 4 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 4 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 4 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1 = 2 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4 = 5 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4 = 5 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4 = 5 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4 = 5 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4 = 5 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4 = 5 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1 = 2 - 1
[1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,4,5,7,3,6] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,4,5,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 2 - 1
[1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,4,6,3,7,5] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
[1,2,4,6,7,3,5] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,6,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,7,3,5,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,7,3,6,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,7,5,3,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,7,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,7,6,3,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 2 - 1
[1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2 - 1
[1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2 - 1
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001645The pebbling number of a connected graph. St000740The last entry of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000051The size of the left subtree of a binary tree. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix.
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