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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
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Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1
[[1,2]]
=> [2] => 2
[[1],[2]]
=> [1,1] => 1
[[1,2,3]]
=> [3] => 3
[[1,3],[2]]
=> [1,2] => 1
[[1,2],[3]]
=> [2,1] => 2
[[1],[2],[3]]
=> [1,1,1] => 1
[[1,2,3,4]]
=> [4] => 4
[[1,3,4],[2]]
=> [1,3] => 1
[[1,2,4],[3]]
=> [2,2] => 2
[[1,2,3],[4]]
=> [3,1] => 3
[[1,3],[2,4]]
=> [1,2,1] => 1
[[1,2],[3,4]]
=> [2,2] => 2
[[1,4],[2],[3]]
=> [1,1,2] => 1
[[1,3],[2],[4]]
=> [1,2,1] => 1
[[1,2],[3],[4]]
=> [2,1,1] => 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[[1,2,3,4,5]]
=> [5] => 5
[[1,3,4,5],[2]]
=> [1,4] => 1
[[1,2,4,5],[3]]
=> [2,3] => 2
[[1,2,3,5],[4]]
=> [3,2] => 3
[[1,2,3,4],[5]]
=> [4,1] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => 1
[[1,2,5],[3,4]]
=> [2,3] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => 1
[[1,2,4],[3,5]]
=> [2,2,1] => 2
[[1,2,3],[4,5]]
=> [3,2] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => 1
[[1,3,5],[2],[4]]
=> [1,2,2] => 1
[[1,2,5],[3],[4]]
=> [2,1,2] => 2
[[1,3,4],[2],[5]]
=> [1,3,1] => 1
[[1,2,4],[3],[5]]
=> [2,2,1] => 2
[[1,2,3],[4],[5]]
=> [3,1,1] => 3
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 1
[[1,3],[2,5],[4]]
=> [1,2,2] => 1
[[1,2],[3,5],[4]]
=> [2,1,2] => 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 1
[[1,2],[3,4],[5]]
=> [2,2,1] => 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[[1,2,3,4,5,6]]
=> [6] => 6
[[1,3,4,5,6],[2]]
=> [1,5] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => 1
Description
The first part of an integer composition.
Matching statistic: St000326
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(load all 3 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
St000326: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => ? = 1
[[1,2]]
=> 0 => 2
[[1],[2]]
=> 1 => 1
[[1,2,3]]
=> 00 => 3
[[1,3],[2]]
=> 10 => 1
[[1,2],[3]]
=> 01 => 2
[[1],[2],[3]]
=> 11 => 1
[[1,2,3,4]]
=> 000 => 4
[[1,3,4],[2]]
=> 100 => 1
[[1,2,4],[3]]
=> 010 => 2
[[1,2,3],[4]]
=> 001 => 3
[[1,3],[2,4]]
=> 101 => 1
[[1,2],[3,4]]
=> 010 => 2
[[1,4],[2],[3]]
=> 110 => 1
[[1,3],[2],[4]]
=> 101 => 1
[[1,2],[3],[4]]
=> 011 => 2
[[1],[2],[3],[4]]
=> 111 => 1
[[1,2,3,4,5]]
=> 0000 => 5
[[1,3,4,5],[2]]
=> 1000 => 1
[[1,2,4,5],[3]]
=> 0100 => 2
[[1,2,3,5],[4]]
=> 0010 => 3
[[1,2,3,4],[5]]
=> 0001 => 4
[[1,3,5],[2,4]]
=> 1010 => 1
[[1,2,5],[3,4]]
=> 0100 => 2
[[1,3,4],[2,5]]
=> 1001 => 1
[[1,2,4],[3,5]]
=> 0101 => 2
[[1,2,3],[4,5]]
=> 0010 => 3
[[1,4,5],[2],[3]]
=> 1100 => 1
[[1,3,5],[2],[4]]
=> 1010 => 1
[[1,2,5],[3],[4]]
=> 0110 => 2
[[1,3,4],[2],[5]]
=> 1001 => 1
[[1,2,4],[3],[5]]
=> 0101 => 2
[[1,2,3],[4],[5]]
=> 0011 => 3
[[1,4],[2,5],[3]]
=> 1101 => 1
[[1,3],[2,5],[4]]
=> 1010 => 1
[[1,2],[3,5],[4]]
=> 0110 => 2
[[1,3],[2,4],[5]]
=> 1011 => 1
[[1,2],[3,4],[5]]
=> 0101 => 2
[[1,5],[2],[3],[4]]
=> 1110 => 1
[[1,4],[2],[3],[5]]
=> 1101 => 1
[[1,3],[2],[4],[5]]
=> 1011 => 1
[[1,2],[3],[4],[5]]
=> 0111 => 2
[[1],[2],[3],[4],[5]]
=> 1111 => 1
[[1,2,3,4,5,6]]
=> 00000 => 6
[[1,3,4,5,6],[2]]
=> 10000 => 1
[[1,2,4,5,6],[3]]
=> 01000 => 2
[[1,2,3,5,6],[4]]
=> 00100 => 3
[[1,2,3,4,6],[5]]
=> 00010 => 4
[[1,2,3,4,5],[6]]
=> 00001 => 5
[[1,3,5,6],[2,4]]
=> 10100 => 1
[[1,2,5,6],[3,4]]
=> 01000 => 2
[[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> 00000100010 => ? = 6
[[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> 00001000101 => ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> 00001001001 => ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 00001001010 => ? = 5
[[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> 00010010101 => ? = 4
[[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> 00010101010 => ? = 4
[[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> 00100101011 => ? = 3
[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> 11111111111 => ? = 1
[[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> 00001000100101 => ? = 5
[[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> 0100100100 => ? = 2
[[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> 11000010000 => ? = 1
[[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> 11010100100 => ? = 1
[[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? = 1
[[1,2],[3,5],[4,6],[7,9],[8,10],[11,12]]
=> 01101110110 => ? = 2
[[1,2],[3,6],[4,7],[5,10],[8,11],[9,12]]
=> 01110111011 => ? = 2
[[1,2],[3,7],[4,8],[5,9],[6,10],[11,12]]
=> 01111011110 => ? = 2
[[1,2],[3,8],[4,9],[5,10],[6,11],[7,12]]
=> 01111101111 => ? = 2
[[1,4],[2,5],[3,6],[7,10],[8,11],[9,12]]
=> 11011111011 => ? = 1
[[1,4],[2,5],[3,7],[6,9],[8,10],[11,12]]
=> 11011010110 => ? = 1
[[1,4],[2,5],[3,8],[6,9],[7,10],[11,12]]
=> 11011101110 => ? = 1
[[1,4],[2,5],[3,8],[6,10],[7,11],[9,12]]
=> 11011101011 => ? = 1
[[1,4],[2,6],[3,7],[5,10],[8,11],[9,12]]
=> 11010111011 => ? = 1
[[1,4],[2,7],[3,8],[5,9],[6,10],[11,12]]
=> 11011011110 => ? = 1
[[1,4],[2,8],[3,9],[5,10],[6,11],[7,12]]
=> 11011101111 => ? = 1
[[1,6],[2,7],[3,8],[4,9],[5,10],[11,12]]
=> 11110111110 => ? = 1
[[1,6],[2,7],[3,8],[4,10],[5,11],[9,12]]
=> 11110111011 => ? = 1
[[1,6],[2,8],[3,9],[4,10],[5,11],[7,12]]
=> 11110101111 => ? = 1
[[1,2,3,4,6,8,10,12],[5,7,9,11]]
=> 00010101010 => ? = 4
[[1,2,3,4,5,6,8,11,12],[7,9,10]]
=> 00000101000 => ? = 6
[[1,2,3,4,5,6,9,10,12],[7,8,11]]
=> 00000100010 => ? = 6
[[1,2,3,4,5,6,9,11,12],[7,8,10]]
=> 00000100100 => ? = 6
[[1,2,3,4,5,7,8,10,12],[6,9,11]]
=> 00001001010 => ? = 5
[[1,2,3,4,5,7,9,10,12],[6,8,11]]
=> 00001010010 => ? = 5
[[1,2,3,4,5,7,9,11,12],[6,8,10]]
=> 00001010100 => ? = 5
[[1,2,3,4,5,6,7,10,11,12],[8,9]]
=> 00000010000 => ? = 7
[[1,2,3,4,5,6,8,9,10,12],[7,11]]
=> 00000100010 => ? = 6
[[1,2,3,4,5,6,8,9,11,12],[7,10]]
=> 00000100100 => ? = 6
[[1,2,3,4,5,6,8,10,11,12],[7,9]]
=> 00000101000 => ? = 6
[[1,2,3,4,5,6,7,9,10,11,12],[8]]
=> 00000010000 => ? = 7
[[1,6,10,12],[2,7,11],[3,8],[4,9],[5]]
=> 11110111010 => ? = 1
[[1,6,10],[2,7,11],[3,8,12],[4,9],[5]]
=> 11110111011 => ? = 1
[[1,5,8,11],[2,6,9],[3,7,10],[4]]
=> 1110110110 => ? = 1
[[1,4,7,9,11],[2,5,8,10],[3,6]]
=> 1101101010 => ? = 1
[[1,5,8,11,13],[2,6,9,12],[3,7,10],[4]]
=> 111011011010 => ? = 1
[[1,5,8,10,11],[2,6,9],[3,7],[4]]
=> 1110110100 => ? = 1
[[1,6,8,10,12],[2,7,9,11],[3],[4],[5]]
=> 11110101010 => ? = 1
[[1,5,9,12,13],[2,6,10],[3,7,11],[4,8]]
=> 111011101100 => ? = 1
[[1,6,9,12,13],[2,7,10],[3,8,11],[4],[5]]
=> 111101101100 => ? = 1
[[1,5,9,10,11],[2,6],[3,7],[4,8]]
=> 1110111000 => ? = 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,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000297
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Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> => ? = 1 - 1
[[1,2]]
=> [[1],[2]]
=> 1 => 1 = 2 - 1
[[1],[2]]
=> [[1,2]]
=> 0 => 0 = 1 - 1
[[1,2,3]]
=> [[1],[2],[3]]
=> 11 => 2 = 3 - 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 01 => 0 = 1 - 1
[[1,2],[3]]
=> [[1,3],[2]]
=> 10 => 1 = 2 - 1
[[1],[2],[3]]
=> [[1,2,3]]
=> 00 => 0 = 1 - 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 111 => 3 = 4 - 1
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 011 => 0 = 1 - 1
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 101 => 1 = 2 - 1
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 110 => 2 = 3 - 1
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 010 => 0 = 1 - 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 101 => 1 = 2 - 1
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 0 = 1 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 0 = 1 - 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 100 => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 000 => 0 = 1 - 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 4 = 5 - 1
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0 = 1 - 1
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 1011 => 1 = 2 - 1
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 1101 => 2 = 3 - 1
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 4 - 1
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 0 = 1 - 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1011 => 1 = 2 - 1
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 0110 => 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 1010 => 1 = 2 - 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 1101 => 2 = 3 - 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0011 => 0 = 1 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 0101 => 0 = 1 - 1
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 1001 => 1 = 2 - 1
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1010 => 1 = 2 - 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 3 - 1
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 0010 => 0 = 1 - 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1001 => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 0100 => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1010 => 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0001 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 0010 => 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 0100 => 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1000 => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0000 => 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 11111 => 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 01111 => 0 = 1 - 1
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 10111 => 1 = 2 - 1
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 11011 => 2 = 3 - 1
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 11101 => 3 = 4 - 1
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 01011 => 0 = 1 - 1
[[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 10111 => 1 = 2 - 1
[[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [[1,2],[3,4],[5,6],[7,8],[9,11],[10,12]]
=> 01010101101 => ? = 1 - 1
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10],[11,12]]
=> 01010110110 => ? = 1 - 1
[[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> [[1,2],[3,4],[5,6],[7,9],[8,11],[10,12]]
=> 01010110101 => ? = 1 - 1
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [[1,2],[3,4],[5,6],[7,10],[8,11],[9,12]]
=> 01010111011 => ? = 1 - 1
[[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10],[11,12]]
=> 01011011010 => ? = 1 - 1
[[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> [[1,2],[3,4],[5,7],[6,8],[9,11],[10,12]]
=> 01011011101 => ? = 1 - 1
[[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10],[11,12]]
=> 01011010110 => ? = 1 - 1
[[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [[1,2],[3,4],[5,7],[6,9],[8,11],[10,12]]
=> 01011010101 => ? = 1 - 1
[[1,3,5,6,8,9],[2,4,7,10,11,12]]
=> [[1,2],[3,4],[5,7],[6,10],[8,11],[9,12]]
=> 01011011011 => ? = 1 - 1
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> [[1,2],[3,4],[5,8],[6,9],[7,10],[11,12]]
=> 01011101110 => ? = 1 - 1
[[1,3,5,6,7,10],[2,4,8,9,11,12]]
=> [[1,2],[3,4],[5,8],[6,9],[7,11],[10,12]]
=> 01011101101 => ? = 1 - 1
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> [[1,2],[3,4],[5,8],[6,10],[7,11],[9,12]]
=> 01011101011 => ? = 1 - 1
[[1,3,5,6,7,8],[2,4,9,10,11,12]]
=> [[1,2],[3,4],[5,9],[6,10],[7,11],[8,12]]
=> 01011110111 => ? = 1 - 1
[[1,3,4,7,9,11],[2,5,6,8,10,12]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10],[11,12]]
=> 01101101010 => ? = 1 - 1
[[1,3,4,7,9,10],[2,5,6,8,11,12]]
=> [[1,2],[3,5],[4,6],[7,8],[9,11],[10,12]]
=> 01101101101 => ? = 1 - 1
[[1,3,4,7,8,11],[2,5,6,9,10,12]]
=> [[1,2],[3,5],[4,6],[7,9],[8,10],[11,12]]
=> 01101110110 => ? = 1 - 1
[[1,3,4,7,8,10],[2,5,6,9,11,12]]
=> [[1,2],[3,5],[4,6],[7,9],[8,11],[10,12]]
=> 01101110101 => ? = 1 - 1
[[1,3,4,7,8,9],[2,5,6,10,11,12]]
=> [[1,2],[3,5],[4,6],[7,10],[8,11],[9,12]]
=> 01101111011 => ? = 1 - 1
[[1,3,4,6,9,11],[2,5,7,8,10,12]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10],[11,12]]
=> 01101011010 => ? = 1 - 1
[[1,3,4,6,9,10],[2,5,7,8,11,12]]
=> [[1,2],[3,5],[4,7],[6,8],[9,11],[10,12]]
=> 01101011101 => ? = 1 - 1
[[1,3,4,6,8,11],[2,5,7,9,10,12]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10],[11,12]]
=> 01101010110 => ? = 1 - 1
[[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> [[1,2],[3,5],[4,7],[6,9],[8,11],[10,12]]
=> 01101010101 => ? = 1 - 1
[[1,3,4,6,8,9],[2,5,7,10,11,12]]
=> [[1,2],[3,5],[4,7],[6,10],[8,11],[9,12]]
=> 01101011011 => ? = 1 - 1
[[1,3,4,6,7,11],[2,5,8,9,10,12]]
=> [[1,2],[3,5],[4,8],[6,9],[7,10],[11,12]]
=> 01101101110 => ? = 1 - 1
[[1,3,4,6,7,10],[2,5,8,9,11,12]]
=> [[1,2],[3,5],[4,8],[6,9],[7,11],[10,12]]
=> 01101101101 => ? = 1 - 1
[[1,3,4,6,7,9],[2,5,8,10,11,12]]
=> [[1,2],[3,5],[4,8],[6,10],[7,11],[9,12]]
=> 01101101011 => ? = 1 - 1
[[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> [[1,2],[3,5],[4,9],[6,10],[7,11],[8,12]]
=> 01101110111 => ? = 1 - 1
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10],[11,12]]
=> 01110111010 => ? = 1 - 1
[[1,3,4,5,9,10],[2,6,7,8,11,12]]
=> [[1,2],[3,6],[4,7],[5,8],[9,11],[10,12]]
=> 01110111101 => ? = 1 - 1
[[1,3,4,5,8,11],[2,6,7,9,10,12]]
=> [[1,2],[3,6],[4,7],[5,9],[8,10],[11,12]]
=> 01110110110 => ? = 1 - 1
[[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> [[1,2],[3,6],[4,7],[5,9],[8,11],[10,12]]
=> 01110110101 => ? = 1 - 1
[[1,3,4,5,8,9],[2,6,7,10,11,12]]
=> [[1,2],[3,6],[4,7],[5,10],[8,11],[9,12]]
=> 01110111011 => ? = 1 - 1
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
=> [[1,2],[3,6],[4,8],[5,9],[7,10],[11,12]]
=> 01110101110 => ? = 1 - 1
[[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> [[1,2],[3,6],[4,8],[5,9],[7,11],[10,12]]
=> 01110101101 => ? = 1 - 1
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
=> [[1,2],[3,6],[4,8],[5,10],[7,11],[9,12]]
=> 01110101011 => ? = 1 - 1
[[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> [[1,2],[3,6],[4,9],[5,10],[7,11],[8,12]]
=> 01110110111 => ? = 1 - 1
[[1,3,4,5,6,11],[2,7,8,9,10,12]]
=> [[1,2],[3,7],[4,8],[5,9],[6,10],[11,12]]
=> 01111011110 => ? = 1 - 1
[[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> [[1,2],[3,7],[4,8],[5,9],[6,11],[10,12]]
=> 01111011101 => ? = 1 - 1
[[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> [[1,2],[3,7],[4,8],[5,10],[6,11],[9,12]]
=> 01111011011 => ? = 1 - 1
[[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> [[1,2],[3,7],[4,9],[5,10],[6,11],[8,12]]
=> 01111010111 => ? = 1 - 1
[[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> [[1,2],[3,8],[4,9],[5,10],[6,11],[7,12]]
=> 01111101111 => ? = 1 - 1
[[1,2,5,7,9,11],[3,4,6,8,10,12]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10],[11,12]]
=> 10110101010 => ? = 2 - 1
[[1,2,5,7,9,10],[3,4,6,8,11,12]]
=> [[1,3],[2,4],[5,6],[7,8],[9,11],[10,12]]
=> 10110101101 => ? = 2 - 1
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10],[11,12]]
=> 10110110110 => ? = 2 - 1
[[1,2,5,7,8,10],[3,4,6,9,11,12]]
=> [[1,3],[2,4],[5,6],[7,9],[8,11],[10,12]]
=> 10110110101 => ? = 2 - 1
[[1,2,5,7,8,9],[3,4,6,10,11,12]]
=> [[1,3],[2,4],[5,6],[7,10],[8,11],[9,12]]
=> 10110111011 => ? = 2 - 1
[[1,2,5,6,9,11],[3,4,7,8,10,12]]
=> [[1,3],[2,4],[5,7],[6,8],[9,10],[11,12]]
=> 10111011010 => ? = 2 - 1
[[1,2,5,6,9,10],[3,4,7,8,11,12]]
=> [[1,3],[2,4],[5,7],[6,8],[9,11],[10,12]]
=> 10111011101 => ? = 2 - 1
[[1,2,5,6,8,11],[3,4,7,9,10,12]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10],[11,12]]
=> 10111010110 => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000745
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
St000745: Standard tableaux ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> 1
[[1,2]]
=> [[1],[2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> 1
[[1,2,3]]
=> [[1],[2],[3]]
=> 3
[[1,3],[2]]
=> [[1,2],[3]]
=> 1
[[1,2],[3]]
=> [[1,3],[2]]
=> 2
[[1],[2],[3]]
=> [[1,2,3]]
=> 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 3
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 2
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 2
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 4
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 1
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 2
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 2
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 2
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 3
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 4
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 5
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 1
[[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8]]
=> ? = 4
[[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9]]
=> ? = 3
[[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 6
[[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> [[1,6,10],[2,7,11],[3,8,12],[4,9],[5]]
=> ? = 5
[[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [[1,6,10,12],[2,7,11],[3,8],[4,9],[5]]
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,6,9,12],[2,7,10],[3,8,11],[4],[5]]
=> ? = 5
[[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,6,9,11],[2,7,10,12],[3,8],[4],[5]]
=> ? = 5
[[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [[1,5,9,12],[2,6,10],[3,7,11],[4,8]]
=> ? = 4
[[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [[1,5,9,11],[2,6,10,12],[3,7],[4,8]]
=> ? = 4
[[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [[1,5,8,11],[2,6,9,12],[3,7,10],[4]]
=> ? = 4
[[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [[1,5,8,10,12],[2,6,9,11],[3,7],[4]]
=> ? = 4
[[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,5,7,9,11],[2,6,8,10,12],[3],[4]]
=> ? = 4
[[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [[1,4,7,10,12],[2,5,8,11],[3,6,9]]
=> ? = 3
[[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [[1,4,7,9,11,12],[2,5,8,10],[3,6]]
=> ? = 3
[[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,6,10,13,15],[2,7,11,14],[3,8,12],[4,9],[5]]
=> ? = 5
[[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 3
[[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 2
[[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 1
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
=> ? = 3
[[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
=> ? = 1
[[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 2
[[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 1
[[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
=> ? = 1
[[1,8],[2,9],[3],[4],[5],[6],[7]]
=> ?
=> ? = 1
[[1,7],[2,8],[3,9],[4],[5],[6]]
=> ?
=> ? = 1
[[1,7,9],[2,8],[3],[4],[5],[6]]
=> ?
=> ? = 1
[[1,6,9],[2,7],[3,8],[4],[5]]
=> ?
=> ? = 1
[[1,6,8],[2,7,9],[3],[4],[5]]
=> ?
=> ? = 1
[[1,5,8],[2,6,9],[3,7],[4]]
=> ?
=> ? = 1
[[1,5,7,9],[2,6,8],[3],[4]]
=> ?
=> ? = 1
[[1,4,7,9],[2,5,8],[3,6]]
=> ?
=> ? = 1
[[1,4,6,8,9],[2,5,7],[3]]
=> ?
=> ? = 1
[[1,8],[2,9],[3,10],[4],[5],[6],[7]]
=> ?
=> ? = 1
[[1,8,10],[2,9],[3],[4],[5],[6],[7]]
=> ?
=> ? = 1
[[1,7,10],[2,8],[3,9],[4],[5],[6]]
=> ?
=> ? = 1
[[1,6,10],[2,7],[3,8],[4,9],[5]]
=> ?
=> ? = 1
[[1,6,8,10],[2,7,9],[3],[4],[5]]
=> ?
=> ? = 1
[[1,4,7,9,10],[2,5,8],[3,6]]
=> ?
=> ? = 1
[[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 1
[[1,4],[2,5],[3,6],[7],[8],[9]]
=> ?
=> ? = 1
[[1,5],[2,6],[3,7],[4,8],[9]]
=> ?
=> ? = 1
[[1,3,5],[2,4,6],[7],[8],[9]]
=> ?
=> ? = 1
[[1,3,6],[2,4,7],[5,8],[9]]
=> ?
=> ? = 1
[[1,3,5,7],[2,4,6,8],[9]]
=> ?
=> ? = 1
[[1,4],[2,5],[3,6],[7],[8],[9],[10]]
=> ?
=> ? = 1
[[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ?
=> ? = 2
[[1,2],[3,4],[5,8],[6,9],[7,10]]
=> ?
=> ? = 2
[[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ?
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000383
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 90%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 90%
Values
[[1]]
=> [1] => [1] => 1
[[1,2]]
=> [2] => [2] => 2
[[1],[2]]
=> [1,1] => [1,1] => 1
[[1,2,3]]
=> [3] => [3] => 3
[[1,3],[2]]
=> [1,2] => [2,1] => 1
[[1,2],[3]]
=> [2,1] => [1,2] => 2
[[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 1
[[1,2,3,4]]
=> [4] => [4] => 4
[[1,3,4],[2]]
=> [1,3] => [3,1] => 1
[[1,2,4],[3]]
=> [2,2] => [2,2] => 2
[[1,2,3],[4]]
=> [3,1] => [1,3] => 3
[[1,3],[2,4]]
=> [1,2,1] => [1,2,1] => 1
[[1,2],[3,4]]
=> [2,2] => [2,2] => 2
[[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,2,1] => 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,2] => 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 1
[[1,2,3,4,5]]
=> [5] => [5] => 5
[[1,3,4,5],[2]]
=> [1,4] => [4,1] => 1
[[1,2,4,5],[3]]
=> [2,3] => [3,2] => 2
[[1,2,3,5],[4]]
=> [3,2] => [2,3] => 3
[[1,2,3,4],[5]]
=> [4,1] => [1,4] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => 1
[[1,2,5],[3,4]]
=> [2,3] => [3,2] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,3,1] => 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,2,2] => 2
[[1,2,3],[4,5]]
=> [3,2] => [2,3] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [2,2,1] => 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [2,1,2] => 2
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,3,1] => 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,2,2] => 2
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,3] => 3
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,2,1,1] => 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [2,1,2] => 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,2,2] => 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [2,1,1,1] => 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,2,1,1] => 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,1,2] => 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2,3,4,5,6]]
=> [6] => [6] => 6
[[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => [4,2] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => [2,4] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,5] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,3,4,5,6,7,8]]
=> [8] => [8] => ? = 8
[[1,2,4,5,6,7,8],[3]]
=> [2,6] => [6,2] => ? = 2
[[1,2,3,5,6,7,8],[4]]
=> [3,5] => [5,3] => ? = 3
[[1,2,3,4,5,7,8],[6]]
=> [5,3] => [3,5] => ? = 5
[[1,2,3,4,5,6,8],[7]]
=> [6,2] => [2,6] => ? = 6
[[1,2,5,6,7,8],[3,4]]
=> [2,6] => [6,2] => ? = 2
[[1,2,4,6,7,8],[3,5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,6,7,8],[4,5]]
=> [3,5] => [5,3] => ? = 3
[[1,3,4,5,6,8],[2,7]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,3,4,6,8],[5,7]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,3,4,5,8],[6,7]]
=> [5,3] => [3,5] => ? = 5
[[1,2,3,4,6,7],[5,8]]
=> [4,3,1] => [1,3,4] => ? = 4
[[1,2,3,4,5,7],[6,8]]
=> [5,2,1] => [1,2,5] => ? = 5
[[1,2,3,4,5,6],[7,8]]
=> [6,2] => [2,6] => ? = 6
[[1,2,5,6,7,8],[3],[4]]
=> [2,1,5] => [5,1,2] => ? = 2
[[1,2,4,6,7,8],[3],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,6,7,8],[4],[5]]
=> [3,1,4] => [4,1,3] => ? = 3
[[1,2,3,4,7,8],[5],[6]]
=> [4,1,3] => [3,1,4] => ? = 4
[[1,3,4,5,6,8],[2],[7]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,3,4,6,8],[5],[7]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,3,4,5,8],[6],[7]]
=> [5,1,2] => [2,1,5] => ? = 5
[[1,2,3,4,6,7],[5],[8]]
=> [4,3,1] => [1,3,4] => ? = 4
[[1,2,3,4,5,7],[6],[8]]
=> [5,2,1] => [1,2,5] => ? = 5
[[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,1,6] => ? = 6
[[1,2,4,7,8],[3,5,6]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,7,8],[4,5,6]]
=> [3,5] => [5,3] => ? = 3
[[1,2,3,4,7],[5,6,8]]
=> [4,3,1] => [1,3,4] => ? = 4
[[1,3,4,5,6],[2,7,8]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,3,4,6],[5,7,8]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,3,4,5],[6,7,8]]
=> [5,3] => [3,5] => ? = 5
[[1,4,6,7,8],[2,5],[3]]
=> [1,1,2,4] => [4,2,1,1] => ? = 1
[[1,2,6,7,8],[3,5],[4]]
=> [2,1,5] => [5,1,2] => ? = 2
[[1,2,6,7,8],[3,4],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,4,7,8],[3,6],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,7,8],[4,6],[5]]
=> [3,1,4] => [4,1,3] => ? = 3
[[1,4,5,6,8],[2,7],[3]]
=> [1,1,4,2] => [2,4,1,1] => ? = 1
[[1,2,3,4,8],[5,7],[6]]
=> [4,1,3] => [3,1,4] => ? = 4
[[1,2,3,4,8],[5,6],[7]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,3,4,5,6],[2,8],[7]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,3,4,6],[5,8],[7]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,3,4,5],[6,8],[7]]
=> [5,1,2] => [2,1,5] => ? = 5
[[1,2,3,4,7],[5,6],[8]]
=> [4,3,1] => [1,3,4] => ? = 4
[[1,2,4,5,6],[3,7],[8]]
=> [2,4,1,1] => [1,1,4,2] => ? = 2
[[1,2,3,4,6],[5,7],[8]]
=> [4,2,1,1] => [1,1,2,4] => ? = 4
[[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,2,5] => ? = 5
[[1,4,6,7,8],[2],[3],[5]]
=> [1,1,2,4] => [4,2,1,1] => ? = 1
[[1,2,6,7,8],[3],[4],[5]]
=> [2,1,1,4] => [4,1,1,2] => ? = 2
[[1,4,5,6,8],[2],[3],[7]]
=> [1,1,4,2] => [2,4,1,1] => ? = 1
[[1,2,3,4,8],[5],[6],[7]]
=> [4,1,1,2] => [2,1,1,4] => ? = 4
[[1,2,4,5,6],[3],[7],[8]]
=> [2,4,1,1] => [1,1,4,2] => ? = 2
Description
The last part of an integer composition.
Matching statistic: St000439
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> 2 = 1 + 1
[[1,2]]
=> [2] => [1,1,0,0]
=> 3 = 2 + 1
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,3,5,6,7,8],[4]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,2,3,4,5,7,8],[6]]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[[1,2,3,6,7,8],[4,5]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,2,3,5,7,8],[4,6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,5,6,8],[4,7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,4,5,8],[6,7]]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[[1,2,3,4,6,7],[5,8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,4,5,7],[6,8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[1,2,3,6,7,8],[4],[5]]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[1,2,3,5,7,8],[4],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,4,7,8],[5],[6]]
=> [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[1,2,3,5,6,8],[4],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,4,5,8],[6],[7]]
=> [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[1,2,3,4,6,7],[5],[8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,4,5,7],[6],[8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,2,3,7,8],[4,5,6]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,2,3,6,8],[4,5,7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5,8],[4,6,7]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,5,7],[4,6,8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,3,4,7],[5,6,8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,5,6],[4,7,8]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[[1,2,3,7,8],[4,6],[5]]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[1,2,3,7,8],[4,5],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,6,8],[4,7],[5]]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5,8],[4,7],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,4,8],[5,7],[6]]
=> [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[1,2,3,6,8],[4,5],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5,8],[4,6],[7]]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,6,7],[4,8],[5]]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,2,3,5,7],[4,8],[6]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,3,4,7],[5,8],[6]]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,5,6],[4,8],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,4,5],[6,8],[7]]
=> [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[1,2,3,5,7],[4,6],[8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,3,4,7],[5,6],[8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,5,6],[4,7],[8]]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,2,3,4,6],[5,7],[8]]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[1,2,3,7,8],[4],[5],[6]]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,6,8],[4],[5],[7]]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5,8],[4],[6],[7]]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,4,8],[5],[6],[7]]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 + 1
[[1,2,3,6,7],[4],[5],[8]]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,2,3,5,7],[4],[6],[8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,3,4,7],[5],[6],[8]]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,5,6],[4],[7],[8]]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,2,3,4,6],[5],[7],[8]]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000678
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 80%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 80%
Values
[[1]]
=> [1] => [1] => [1,0]
=> ? = 1
[[1,2]]
=> [2] => [1,1] => [1,0,1,0]
=> 2
[[1],[2]]
=> [1,1] => [2] => [1,1,0,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[1,3,4],[2]]
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2],[3,4]]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,3],[2],[4]]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2],[3],[4]]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,4,5],[2]]
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4,5]]
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4],[5]]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5,6],[2]]
=> [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,5,6],[3,4]]
=> [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,4,5,6],[2,7],[8]]
=> [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,5,6],[3,7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,6],[4,7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,6],[5,7],[8]]
=> [4,2,1,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,3,4,5,6],[2],[7],[8]]
=> [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,5,6],[3],[7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,6],[4],[7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,6],[5],[7],[8]]
=> [4,2,1,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[1,3,5,6],[2,4,7],[8]]
=> [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,5,6],[3,4,7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,3,4,6],[2,5,7],[8]]
=> [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,6],[3,5,7],[8]]
=> [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,6],[4,5,7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,4,5,6],[2,7],[3],[8]]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,5,6],[2,7],[4],[8]]
=> [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,5,6],[3,7],[4],[8]]
=> [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,3,4,6],[2,7],[5],[8]]
=> [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,6],[3,7],[5],[8]]
=> [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,6],[4,7],[5],[8]]
=> [3,1,2,1,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,5,6],[2,4],[7],[8]]
=> [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,5,6],[3,4],[7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,3,4,6],[2,5],[7],[8]]
=> [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,6],[3,5],[7],[8]]
=> [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,6],[4,5],[7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,3,4,5],[2,6],[7],[8]]
=> [1,4,1,1,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,5],[3,6],[7],[8]]
=> [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,5],[4,6],[7],[8]]
=> [3,2,1,1,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,4,5,6],[2],[3],[7],[8]]
=> [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,5,6],[2],[4],[7],[8]]
=> [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,5,6],[3],[4],[7],[8]]
=> [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,3,4,6],[2],[5],[7],[8]]
=> [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,6],[3],[5],[7],[8]]
=> [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,6],[4],[5],[7],[8]]
=> [3,1,2,1,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,4,5],[2],[6],[7],[8]]
=> [1,4,1,1,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4,5],[3],[6],[7],[8]]
=> [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,3,5],[4],[6],[7],[8]]
=> [3,2,1,1,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,4,6],[2,5,7],[3],[8]]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,6],[2,5,7],[4],[8]]
=> [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,6],[3,5,7],[4],[8]]
=> [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,3,6],[2,4,7],[5],[8]]
=> [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,6],[3,4,7],[5],[8]]
=> [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,3,5],[2,4,6],[7],[8]]
=> [1,2,2,1,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,5],[3,4,6],[7],[8]]
=> [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,3,4],[2,5,6],[7],[8]]
=> [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,4],[3,5,6],[7],[8]]
=> [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000011
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,0]
=> 1
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[[1,3,4,6,7,8],[2,5]]
=> [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,6,7,8],[3,5]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,6,8],[4,7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,6,8],[5,7]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,3,4,6,7,8],[2],[5]]
=> [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,6,7,8],[3],[5]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,6,8],[4],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,6,8],[5],[7]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,3,5,7,8],[2,4,6]]
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[1,3,4,7,8],[2,5,6]]
=> [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,7,8],[3,5,6]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 1
[[1,3,4,6,8],[2,5,7]]
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,6,8],[4,5,7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,3,4,6,7],[2,5,8]]
=> [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,6,7],[3,5,8]]
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,5,7],[3,6,8]]
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,5,6],[4,7,8]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,6],[5,7,8]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,2,6,7,8],[3,4],[5]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,5,7,8],[2,6],[4]]
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[1,2,5,7,8],[3,6],[4]]
=> [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2
[[1,3,4,7,8],[2,6],[5]]
=> [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,7,8],[3,6],[5]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,5,7,8],[2,4],[6]]
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[[1,2,4,7,8],[3,5],[6]]
=> [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,5,6,8],[2,7],[4]]
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 1
[[1,2,5,6,8],[3,7],[4]]
=> [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[[1,3,4,6,8],[2,7],[5]]
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,6,8],[4,7],[5]]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,5,6,8],[2,4],[7]]
=> [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 1
[[1,3,4,6,8],[2,5],[7]]
=> [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,6,8],[4,5],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,4,5,8],[3,6],[7]]
=> [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,8],[4,6],[7]]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,8],[5,6],[7]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,2,5,6,7],[3,8],[4]]
=> [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[[1,3,4,6,7],[2,8],[5]]
=> [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,6,7],[3,8],[5]]
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,6,7],[4,8],[5]]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,4,5,7],[2,8],[6]]
=> [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,5,7],[3,8],[6]]
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,7],[4,8],[6]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,5,6],[4,8],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[[1,2,3,4,6],[5,8],[7]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,3,4,6,7],[2,5],[8]]
=> [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,4,6,7],[3,5],[8]]
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,5,7],[2,6],[8]]
=> [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000617
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 90%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 90%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 1
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 2
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 3
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 2
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [3,4,1,5,2,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [4,5,1,2,6,3,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [4,1,5,6,2,3,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [4,5,1,6,7,2,3] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [4,2,1,5,3,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [4,2,5,1,3,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [4,5,2,1,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [4,2,5,3,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [4,5,2,3,1,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [4,2,1,5,6,3,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [4,5,2,1,6,3,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [4,2,5,6,1,3,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,2,5,7],[3,4],[6]]
=> [6,3,4,1,2,5,7] => [4,5,2,3,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,7],[2,5],[6]]
=> [6,2,5,1,3,4,7] => [4,2,5,6,3,1,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [4,2,1,5,6,7,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [4,2,5,1,6,7,3] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => [4,5,2,1,6,7,3] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [4,5,2,6,7,1,3] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [4,5,6,2,7,1,3] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [4,2,5,3,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3,4],[7]]
=> [7,3,4,1,2,5,6] => [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[1,2,4,5],[3,6],[7]]
=> [7,3,6,1,2,4,5] => [4,5,2,6,7,3,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => [4,5,6,2,7,3,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [4,5,3,2,1,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [4,5,3,2,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [4,3,5,6,2,1,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [5,2,1,6,7,3,4] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [5,2,6,7,1,3,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [5,2,6,7,3,1,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,3,4],[2,5,6],[7]]
=> [7,2,5,6,1,3,4] => [5,2,6,7,3,4,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [5,3,1,6,7,4,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [5,3,6,7,1,4,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St000054
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1] => 1
[[1,2]]
=> [2] => [1,1,0,0]
=> [2,1] => 2
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [1,2] => 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [3,1,2] => 3
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 6
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => 1
[[1,2,3,5,6,7],[4]]
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 3
[[1,2,3,4,6,7],[5]]
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4
[[1,3,5,6,7],[2,4]]
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 1
[[1,3,4,6,7],[2,5]]
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 1
[[1,2,4,6,7],[3,5]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 2
[[1,2,3,6,7],[4,5]]
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 3
[[1,2,4,5,7],[3,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 2
[[1,2,3,5,7],[4,6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 3
[[1,2,3,4,7],[5,6]]
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4
[[1,3,4,5,6],[2,7]]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3
[[1,2,3,4,6],[5,7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 4
[[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1
[[1,3,5,6,7],[2],[4]]
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 1
[[1,3,4,6,7],[2],[5]]
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 1
[[1,2,4,6,7],[3],[5]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 2
[[1,2,3,6,7],[4],[5]]
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 3
[[1,2,4,5,7],[3],[6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 2
[[1,2,3,5,7],[4],[6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 3
[[1,3,4,5,6],[2],[7]]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 1
[[1,2,3,5,6],[4],[7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3
[[1,2,3,4,6],[5],[7]]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 4
[[1,3,5,7],[2,4,6]]
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 1
[[1,2,5,7],[3,4,6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 2
[[1,3,4,7],[2,5,6]]
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 1
[[1,2,4,7],[3,5,6]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 2
[[1,2,3,7],[4,5,6]]
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 3
[[1,3,5,6],[2,4,7]]
=> [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => ? = 1
[[1,3,4,6],[2,5,7]]
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,3,6,5,7] => ? = 1
[[1,2,4,6],[3,5,7]]
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3
[[1,2,4,5],[3,6,7]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 3
[[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4
[[1,4,6,7],[2,5],[3]]
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => ? = 1
[[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => ? = 1
[[1,3,6,7],[2,4],[5]]
=> [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => ? = 1
[[1,2,6,7],[3,4],[5]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 2
[[1,4,5,7],[2,6],[3]]
=> [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => ? = 1
[[1,3,5,7],[2,6],[4]]
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 1
[[1,2,5,7],[3,6],[4]]
=> [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 2
[[1,3,4,7],[2,6],[5]]
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,2,3,7,5,6] => ? = 1
[[1,2,4,7],[3,6],[5]]
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 2
[[1,2,3,7],[4,6],[5]]
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 3
[[1,3,5,7],[2,4],[6]]
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 1
[[1,2,5,7],[3,4],[6]]
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 2
[[1,3,4,7],[2,5],[6]]
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,2,3,5,7,6] => ? = 1
[[1,2,4,7],[3,5],[6]]
=> [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 2
[[1,2,3,7],[4,5],[6]]
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 3
[[1,4,5,6],[2,7],[3]]
=> [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ? = 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 π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. 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).
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001050The number of terminal closers of a set partition. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001733The number of weak left to right maxima of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001316The domatic number of a graph. St000234The number of global ascents of a permutation. St000989The number of final rises 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. St000654The first descent of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000546The number of global descents of a permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000260The radius of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation.
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