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Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
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(load all 7 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 => 1
[[1,2]]
=> [2] => 10 => 1
[[1],[2]]
=> [1,1] => 11 => 2
[[1,2,3]]
=> [3] => 100 => 1
[[1,3],[2]]
=> [1,2] => 110 => 2
[[1,2],[3]]
=> [2,1] => 101 => 1
[[1],[2],[3]]
=> [1,1,1] => 111 => 3
[[1,2,3,4]]
=> [4] => 1000 => 1
[[1,3,4],[2]]
=> [1,3] => 1100 => 2
[[1,2,4],[3]]
=> [2,2] => 1010 => 1
[[1,2,3],[4]]
=> [3,1] => 1001 => 1
[[1,3],[2,4]]
=> [1,2,1] => 1101 => 2
[[1,2],[3,4]]
=> [2,2] => 1010 => 1
[[1,4],[2],[3]]
=> [1,1,2] => 1110 => 3
[[1,3],[2],[4]]
=> [1,2,1] => 1101 => 2
[[1,2],[3],[4]]
=> [2,1,1] => 1011 => 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1111 => 4
[[1,2,3,4,5]]
=> [5] => 10000 => 1
[[1,3,4,5],[2]]
=> [1,4] => 11000 => 2
[[1,2,4,5],[3]]
=> [2,3] => 10100 => 1
[[1,2,3,5],[4]]
=> [3,2] => 10010 => 1
[[1,2,3,4],[5]]
=> [4,1] => 10001 => 1
[[1,3,5],[2,4]]
=> [1,2,2] => 11010 => 2
[[1,2,5],[3,4]]
=> [2,3] => 10100 => 1
[[1,3,4],[2,5]]
=> [1,3,1] => 11001 => 2
[[1,2,4],[3,5]]
=> [2,2,1] => 10101 => 1
[[1,2,3],[4,5]]
=> [3,2] => 10010 => 1
[[1,4,5],[2],[3]]
=> [1,1,3] => 11100 => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => 11010 => 2
[[1,2,5],[3],[4]]
=> [2,1,2] => 10110 => 1
[[1,3,4],[2],[5]]
=> [1,3,1] => 11001 => 2
[[1,2,4],[3],[5]]
=> [2,2,1] => 10101 => 1
[[1,2,3],[4],[5]]
=> [3,1,1] => 10011 => 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 11101 => 3
[[1,3],[2,5],[4]]
=> [1,2,2] => 11010 => 2
[[1,2],[3,5],[4]]
=> [2,1,2] => 10110 => 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 11011 => 2
[[1,2],[3,4],[5]]
=> [2,2,1] => 10101 => 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 11110 => 4
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 11101 => 3
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 11011 => 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 10111 => 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 11111 => 5
[[1,2,3,4,5,6]]
=> [6] => 100000 => 1
[[1,3,4,5,6],[2]]
=> [1,5] => 110000 => 2
[[1,2,4,5,6],[3]]
=> [2,4] => 101000 => 1
[[1,2,3,5,6],[4]]
=> [3,3] => 100100 => 1
[[1,2,3,4,6],[5]]
=> [4,2] => 100010 => 1
[[1,2,3,4,5],[6]]
=> [5,1] => 100001 => 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => 110100 => 2
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 => 0 => 2 = 1 + 1
[[1,2]]
=> [2] => 10 => 01 => 2 = 1 + 1
[[1],[2]]
=> [1,1] => 11 => 00 => 3 = 2 + 1
[[1,2,3]]
=> [3] => 100 => 011 => 2 = 1 + 1
[[1,3],[2]]
=> [1,2] => 110 => 001 => 3 = 2 + 1
[[1,2],[3]]
=> [2,1] => 101 => 010 => 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1] => 111 => 000 => 4 = 3 + 1
[[1,2,3,4]]
=> [4] => 1000 => 0111 => 2 = 1 + 1
[[1,3,4],[2]]
=> [1,3] => 1100 => 0011 => 3 = 2 + 1
[[1,2,4],[3]]
=> [2,2] => 1010 => 0101 => 2 = 1 + 1
[[1,2,3],[4]]
=> [3,1] => 1001 => 0110 => 2 = 1 + 1
[[1,3],[2,4]]
=> [1,2,1] => 1101 => 0010 => 3 = 2 + 1
[[1,2],[3,4]]
=> [2,2] => 1010 => 0101 => 2 = 1 + 1
[[1,4],[2],[3]]
=> [1,1,2] => 1110 => 0001 => 4 = 3 + 1
[[1,3],[2],[4]]
=> [1,2,1] => 1101 => 0010 => 3 = 2 + 1
[[1,2],[3],[4]]
=> [2,1,1] => 1011 => 0100 => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1111 => 0000 => 5 = 4 + 1
[[1,2,3,4,5]]
=> [5] => 10000 => 01111 => 2 = 1 + 1
[[1,3,4,5],[2]]
=> [1,4] => 11000 => 00111 => 3 = 2 + 1
[[1,2,4,5],[3]]
=> [2,3] => 10100 => 01011 => 2 = 1 + 1
[[1,2,3,5],[4]]
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
[[1,2,3,4],[5]]
=> [4,1] => 10001 => 01110 => 2 = 1 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => 11010 => 00101 => 3 = 2 + 1
[[1,2,5],[3,4]]
=> [2,3] => 10100 => 01011 => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [1,3,1] => 11001 => 00110 => 3 = 2 + 1
[[1,2,4],[3,5]]
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
[[1,2,3],[4,5]]
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => 11100 => 00011 => 4 = 3 + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => 11010 => 00101 => 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => 10110 => 01001 => 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => 11001 => 00110 => 3 = 2 + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => 10011 => 01100 => 2 = 1 + 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 11101 => 00010 => 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => 11010 => 00101 => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => 10110 => 01001 => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 11011 => 00100 => 3 = 2 + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 11110 => 00001 => 5 = 4 + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 11101 => 00010 => 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 11011 => 00100 => 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 10111 => 01000 => 2 = 1 + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 11111 => 00000 => 6 = 5 + 1
[[1,2,3,4,5,6]]
=> [6] => 100000 => 011111 => 2 = 1 + 1
[[1,3,4,5,6],[2]]
=> [1,5] => 110000 => 001111 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
=> [2,4] => 101000 => 010111 => 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> [3,3] => 100100 => 011011 => 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [4,2] => 100010 => 011101 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [5,1] => 100001 => 011110 => 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => 110100 => 001011 => 3 = 2 + 1
[[1,3,4,7,9],[2,5,6,8,10]]
=> [1,3,3,2,1] => 1100100101 => 0011011010 => ? = 2 + 1
[[1,3,4,6,9],[2,5,7,8,10]]
=> [1,3,2,3,1] => 1100101001 => 0011010110 => ? = 2 + 1
[[1,3,4,5,8],[2,6,7,9,10]]
=> [1,4,3,2] => 1100010010 => 0011101101 => ? = 2 + 1
[[1,3,4,5,6],[2,7,8,9,10]]
=> [1,5,4] => 1100001000 => 0011110111 => ? = 2 + 1
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [1,2,2,3,3,1] => 110101001001 => 001010110110 => ? = 2 + 1
[[1,3,4,6,9,11],[2,5,7,8,10,12]]
=> [1,3,2,3,2,1] => 110010100101 => 001101011010 => ? = 2 + 1
[[1,3,4,6,8,11],[2,5,7,9,10,12]]
=> [1,3,2,2,3,1] => 110010101001 => 001101010110 => ? = 2 + 1
[[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => 10001000100 => 01110111011 => ? = 1 + 1
[[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => 10010010010 => 01101101101 => ? = 1 + 1
[[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [3,4,4] => 10010001000 => 01101110111 => ? = 1 + 1
[[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [2,3,3,3] => 10100100100 => ? => ? = 1 + 1
[[1,2,5,6,9],[3,7,8,10],[4]]
=> [2,1,3,3,1] => 1011001001 => 0100110110 => ? = 1 + 1
[[1,3,4,5,6],[2,8,9,10],[7]]
=> [1,5,4] => 1100001000 => 0011110111 => ? = 2 + 1
[[1,3,4,5],[2,7,8],[6,9,10]]
=> [1,4,3,2] => 1100010010 => 0011101101 => ? = 2 + 1
[[1,3,4,5],[2,7,8],[6,10],[9]]
=> [1,4,3,2] => 1100010010 => 0011101101 => ? = 2 + 1
[[1,3,4],[2,6,7],[5,9],[8,10]]
=> [1,3,3,2,1] => 1100100101 => 0011011010 => ? = 2 + 1
[[1,3,4],[2,6,7],[5,9],[8],[10]]
=> [1,3,3,2,1] => 1100100101 => 0011011010 => ? = 2 + 1
[[1,2,6,7,8,9],[3],[4],[5],[10]]
=> [2,1,1,5,1] => 1011100001 => 0100011110 => ? = 1 + 1
[[1,3,4,6],[2,7,9],[5,8,10]]
=> [1,3,2,3,1] => 1100101001 => 0011010110 => ? = 2 + 1
[[1,3,4,6,9],[2,7,10],[5,8]]
=> [1,3,2,3,1] => 1100101001 => 0011010110 => ? = 2 + 1
[[1,3,4,7,10],[2,5,6,8],[9]]
=> [1,3,3,1,2] => 1100100110 => 0011011001 => ? = 2 + 1
[[1,2,5,6,9],[3,7,8],[4,10]]
=> [2,1,3,3,1] => 1011001001 => 0100110110 => ? = 1 + 1
[[1,3,4,5,8,10],[2,7],[6],[9]]
=> [1,4,3,2] => 1100010010 => 0011101101 => ? = 2 + 1
[[1,3,4,9],[2,6,7,10],[5],[8]]
=> [1,3,3,2,1] => 1100100101 => 0011011010 => ? = 2 + 1
[[1,3,4,9],[2,6,8,10],[5],[7]]
=> [1,3,2,3,1] => 1100101001 => 0011010110 => ? = 2 + 1
[[1,3,4,9],[2,7,8,10],[5],[6]]
=> [1,3,1,4,1] => 1100110001 => 0011001110 => ? = 2 + 1
[[1,3,4,5],[2,7,8,10],[6],[9]]
=> [1,4,3,2] => 1100010010 => 0011101101 => ? = 2 + 1
[[1,3,7,8,11],[2,5,9,10,12],[4],[6]]
=> [1,2,2,3,3,1] => 110101001001 => 001010110110 => ? = 2 + 1
[[1,3,5,9,11],[2,7,8,10,12],[4],[6]]
=> [1,2,2,4,2,1] => 110101000101 => 001010111010 => ? = 2 + 1
[[1,3,5,8,11],[2,7,9,10,12],[4],[6]]
=> [1,2,2,3,3,1] => 110101001001 => 001010110110 => ? = 2 + 1
[[1,3,5,7,11],[2,8,9,10,12],[4],[6]]
=> [1,2,2,2,4,1] => 110101010001 => 001010101110 => ? = 2 + 1
[[1,3,4,9,11],[2,6,8,10,12],[5],[7]]
=> [1,3,2,3,2,1] => 110010100101 => 001101011010 => ? = 2 + 1
[[1,3,4,8,11],[2,6,9,10,12],[5],[7]]
=> [1,3,2,2,3,1] => 110010101001 => 001101010110 => ? = 2 + 1
[[1,3,4,7],[2,5,6,8],[9,10]]
=> [1,3,3,1,2] => 1100100110 => 0011011001 => ? = 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,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000382
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 => [1] => 1
[[1,2]]
=> [2] => 10 => [1,1] => 1
[[1],[2]]
=> [1,1] => 11 => [2] => 2
[[1,2,3]]
=> [3] => 100 => [1,2] => 1
[[1,3],[2]]
=> [1,2] => 110 => [2,1] => 2
[[1,2],[3]]
=> [2,1] => 101 => [1,1,1] => 1
[[1],[2],[3]]
=> [1,1,1] => 111 => [3] => 3
[[1,2,3,4]]
=> [4] => 1000 => [1,3] => 1
[[1,3,4],[2]]
=> [1,3] => 1100 => [2,2] => 2
[[1,2,4],[3]]
=> [2,2] => 1010 => [1,1,1,1] => 1
[[1,2,3],[4]]
=> [3,1] => 1001 => [1,2,1] => 1
[[1,3],[2,4]]
=> [1,2,1] => 1101 => [2,1,1] => 2
[[1,2],[3,4]]
=> [2,2] => 1010 => [1,1,1,1] => 1
[[1,4],[2],[3]]
=> [1,1,2] => 1110 => [3,1] => 3
[[1,3],[2],[4]]
=> [1,2,1] => 1101 => [2,1,1] => 2
[[1,2],[3],[4]]
=> [2,1,1] => 1011 => [1,1,2] => 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1111 => [4] => 4
[[1,2,3,4,5]]
=> [5] => 10000 => [1,4] => 1
[[1,3,4,5],[2]]
=> [1,4] => 11000 => [2,3] => 2
[[1,2,4,5],[3]]
=> [2,3] => 10100 => [1,1,1,2] => 1
[[1,2,3,5],[4]]
=> [3,2] => 10010 => [1,2,1,1] => 1
[[1,2,3,4],[5]]
=> [4,1] => 10001 => [1,3,1] => 1
[[1,3,5],[2,4]]
=> [1,2,2] => 11010 => [2,1,1,1] => 2
[[1,2,5],[3,4]]
=> [2,3] => 10100 => [1,1,1,2] => 1
[[1,3,4],[2,5]]
=> [1,3,1] => 11001 => [2,2,1] => 2
[[1,2,4],[3,5]]
=> [2,2,1] => 10101 => [1,1,1,1,1] => 1
[[1,2,3],[4,5]]
=> [3,2] => 10010 => [1,2,1,1] => 1
[[1,4,5],[2],[3]]
=> [1,1,3] => 11100 => [3,2] => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => 11010 => [2,1,1,1] => 2
[[1,2,5],[3],[4]]
=> [2,1,2] => 10110 => [1,1,2,1] => 1
[[1,3,4],[2],[5]]
=> [1,3,1] => 11001 => [2,2,1] => 2
[[1,2,4],[3],[5]]
=> [2,2,1] => 10101 => [1,1,1,1,1] => 1
[[1,2,3],[4],[5]]
=> [3,1,1] => 10011 => [1,2,2] => 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 11101 => [3,1,1] => 3
[[1,3],[2,5],[4]]
=> [1,2,2] => 11010 => [2,1,1,1] => 2
[[1,2],[3,5],[4]]
=> [2,1,2] => 10110 => [1,1,2,1] => 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 11011 => [2,1,2] => 2
[[1,2],[3,4],[5]]
=> [2,2,1] => 10101 => [1,1,1,1,1] => 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 11110 => [4,1] => 4
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 11101 => [3,1,1] => 3
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 11011 => [2,1,2] => 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 10111 => [1,1,3] => 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 11111 => [5] => 5
[[1,2,3,4,5,6]]
=> [6] => 100000 => [1,5] => 1
[[1,3,4,5,6],[2]]
=> [1,5] => 110000 => [2,4] => 2
[[1,2,4,5,6],[3]]
=> [2,4] => 101000 => [1,1,1,3] => 1
[[1,2,3,5,6],[4]]
=> [3,3] => 100100 => [1,2,1,2] => 1
[[1,2,3,4,6],[5]]
=> [4,2] => 100010 => [1,3,1,1] => 1
[[1,2,3,4,5],[6]]
=> [5,1] => 100001 => [1,4,1] => 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => 110100 => [2,1,1,2] => 2
[[1,3,4,6,9],[2,5,7,8,10]]
=> [1,3,2,3,1] => 1100101001 => [2,2,1,1,1,2,1] => ? = 2
[[1,3,4,5,8],[2,6,7,9,10]]
=> [1,4,3,2] => 1100010010 => [2,3,1,2,1,1] => ? = 2
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [1,2,2,3,3,1] => 110101001001 => [2,1,1,1,1,2,1,2,1] => ? = 2
[[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> [1,2,3,3,2,1] => 110100100101 => [2,1,1,2,1,2,1,1,1] => ? = 2
[[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> [1,2,3,2,3,1] => 110100101001 => [2,1,1,2,1,1,1,2,1] => ? = 2
[[1,3,4,7,9,11],[2,5,6,8,10,12]]
=> [1,3,3,2,2,1] => 110010010101 => [2,2,1,2,1,1,1,1,1] => ? = 2
[[1,3,4,6,9,11],[2,5,7,8,10,12]]
=> [1,3,2,3,2,1] => 110010100101 => [2,2,1,1,1,2,1,1,1] => ? = 2
[[1,3,4,6,8,11],[2,5,7,9,10,12]]
=> [1,3,2,2,3,1] => 110010101001 => [2,2,1,1,1,1,1,2,1] => ? = 2
[[1,2,5,7,9,11],[3,4,6,8,10,12]]
=> [2,3,2,2,2,1] => 101001010101 => [1,1,1,2,1,1,1,1,1,1,1] => ? = 1
[[1,2,4,7,9,11],[3,5,6,8,10,12]]
=> [2,2,3,2,2,1] => 101010010101 => [1,1,1,1,1,2,1,1,1,1,1] => ? = 1
[[1,2,4,6,9,11],[3,5,7,8,10,12]]
=> [2,2,2,3,2,1] => 101010100101 => [1,1,1,1,1,1,1,2,1,1,1] => ? = 1
[[1,2,4,6,8,11],[3,5,7,9,10,12]]
=> [2,2,2,2,3,1] => 101010101001 => [1,1,1,1,1,1,1,1,1,2,1] => ? = 1
[[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => 10001000100 => [1,3,1,3,1,2] => ? = 1
[[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => 10010010010 => [1,2,1,2,1,2,1,1] => ? = 1
[[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [2,3,3,3] => 10100100100 => [1,1,1,2,1,2,1,2] => ? = 1
[[1,2,3,5,8],[4,6,9],[7,10]]
=> [3,2,1,2,2] => 1001011010 => [1,2,1,1,2,1,1,1] => ? = 1
[[1,2,3,10],[4,5,6],[7],[8],[9]]
=> [3,3,1,1,2] => 1001001110 => [1,2,1,2,3,1] => ? = 1
[[1,2,3,10],[4,5],[6,7],[8],[9]]
=> [3,2,2,1,2] => 1001010110 => [1,2,1,1,1,1,2,1] => ? = 1
[[1,2,5],[3,4,10],[6,7],[8],[9]]
=> [2,3,2,1,2] => 1010010110 => [1,1,1,2,1,1,2,1] => ? = 1
[[1,2,5,6,9],[3,7,8,10],[4]]
=> [2,1,3,3,1] => 1011001001 => [1,1,2,2,1,2,1] => ? = 1
[[1,3,4,5],[2,7,8],[6,9,10]]
=> [1,4,3,2] => 1100010010 => [2,3,1,2,1,1] => ? = 2
[[1,3,4,5],[2,7,8],[6,10],[9]]
=> [1,4,3,2] => 1100010010 => [2,3,1,2,1,1] => ? = 2
[[1,3,4,5],[2,7],[6,9],[8,10]]
=> [1,4,2,2,1] => 1100010101 => [2,3,1,1,1,1,1] => ? = 2
[[1,3,4,5],[2,7],[6,9],[8],[10]]
=> [1,4,2,2,1] => 1100010101 => [2,3,1,1,1,1,1] => ? = 2
[[1,4,7,8,9],[2,6],[3],[5],[10]]
=> [1,1,2,5,1] => 1110100001 => [3,1,1,4,1] => ? = 3
[[1,4,5,9],[2,7,8],[3],[6],[10]]
=> [1,1,3,4,1] => 1110010001 => [3,2,1,3,1] => ? = 3
[[1,3,8,9],[2,5],[4,7],[6,10]]
=> [1,2,2,4,1] => 1101010001 => [2,1,1,1,1,3,1] => ? = 2
[[1,3,8,9],[2,5],[4,7],[6],[10]]
=> [1,2,2,4,1] => 1101010001 => [2,1,1,1,1,3,1] => ? = 2
[[1,2],[3,9],[4],[5],[6],[7],[8],[10]]
=> [2,1,1,1,1,1,2,1] => 1011111101 => [1,1,6,1,1] => ? = 1
[[1,2,6,7,8,9],[3],[4],[5],[10]]
=> [2,1,1,5,1] => 1011100001 => [1,1,3,4,1] => ? = 1
[[1,2,4,5],[3,8],[6,9],[7,10]]
=> [2,3,1,2,2] => 1010011010 => [1,1,1,2,2,1,1,1] => ? = 1
[[1,3,4,6],[2,7,9],[5,8,10]]
=> [1,3,2,3,1] => 1100101001 => [2,2,1,1,1,2,1] => ? = 2
[[1,2,9],[3,10],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,2,1] => 1011111101 => [1,1,6,1,1] => ? = 1
[[1,2,3,4],[5,7],[6,8],[9,10]]
=> [4,1,2,1,2] => 1000110110 => [1,3,2,1,2,1] => ? = 1
[[1,2,3,4],[5,8],[6,9],[7,10]]
=> [4,1,1,2,2] => 1000111010 => [1,3,3,1,1,1] => ? = 1
[[1,2,3,5],[4,6],[7,8],[9,10]]
=> [3,2,1,2,2] => 1001011010 => [1,2,1,1,2,1,1,1] => ? = 1
[[1,2,3,5],[4,7],[6,8],[9,10]]
=> [3,2,2,1,2] => 1001010110 => [1,2,1,1,1,1,2,1] => ? = 1
[[1,2,3,5],[4,8],[6,9],[7,10]]
=> [3,2,1,2,2] => 1001011010 => [1,2,1,1,2,1,1,1] => ? = 1
[[1,2,3,6],[4,7],[5,8],[9,10]]
=> [3,1,2,2,2] => 1001101010 => [1,2,2,1,1,1,1,1] => ? = 1
[[1,2,3,6],[4,8],[5,9],[7,10]]
=> [3,1,2,2,2] => 1001101010 => [1,2,2,1,1,1,1,1] => ? = 1
[[1,2,4,5],[3,6],[7,8],[9,10]]
=> [2,3,1,2,2] => 1010011010 => [1,1,1,2,2,1,1,1] => ? = 1
[[1,2,4,5],[3,7],[6,8],[9,10]]
=> [2,3,2,1,2] => 1010010110 => [1,1,1,2,1,1,2,1] => ? = 1
[[1,2,3,5],[4,7],[6,9],[8,11],[10,12]]
=> [3,2,2,2,2,1] => 100101010101 => [1,2,1,1,1,1,1,1,1,1,1] => ? = 1
[[1,2,4,5],[3,7],[6,9],[8,11],[10,12]]
=> [2,3,2,2,2,1] => 101001010101 => [1,1,1,2,1,1,1,1,1,1,1] => ? = 1
[[1,4,7,8,9],[2,5],[3,6],[10]]
=> [1,1,2,5,1] => 1110100001 => [3,1,1,4,1] => ? = 3
[[1,4,6,7,8,9],[2,5],[3],[10]]
=> [1,1,2,5,1] => 1110100001 => [3,1,1,4,1] => ? = 3
[[1,3,6,7,8,9],[2,4],[5,10]]
=> [1,2,1,5,1] => 1101100001 => [2,1,2,4,1] => ? = 2
[[1,3,5,7,8,9],[2,4,6],[10]]
=> [1,2,2,4,1] => 1101010001 => [2,1,1,1,1,3,1] => ? = 2
[[1,2],[3,9],[4,10],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,2,1] => 1011111101 => [1,1,6,1,1] => ? = 1
[[1,2,3,6,10],[4,7],[5,8],[9]]
=> [3,1,2,2,2] => 1001101010 => [1,2,2,1,1,1,1,1] => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000745
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000745: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,2]]
=> 1
[[1],[2]]
=> 2
[[1,2,3]]
=> 1
[[1,3],[2]]
=> 2
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 3
[[1,2,3,4]]
=> 1
[[1,3,4],[2]]
=> 2
[[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 1
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 1
[[1],[2],[3],[4]]
=> 4
[[1,2,3,4,5]]
=> 1
[[1,3,4,5],[2]]
=> 2
[[1,2,4,5],[3]]
=> 1
[[1,2,3,5],[4]]
=> 1
[[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 1
[[1,3,4],[2,5]]
=> 2
[[1,2,4],[3,5]]
=> 1
[[1,2,3],[4,5]]
=> 1
[[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> 1
[[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> 1
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 2
[[1,2],[3,5],[4]]
=> 1
[[1,3],[2,4],[5]]
=> 2
[[1,2],[3,4],[5]]
=> 1
[[1,5],[2],[3],[4]]
=> 4
[[1,4],[2],[3],[5]]
=> 3
[[1,3],[2],[4],[5]]
=> 2
[[1,2],[3],[4],[5]]
=> 1
[[1],[2],[3],[4],[5]]
=> 5
[[1,2,3,4,5,6]]
=> 1
[[1,3,4,5,6],[2]]
=> 2
[[1,2,4,5,6],[3]]
=> 1
[[1,2,3,5,6],[4]]
=> 1
[[1,2,3,4,6],[5]]
=> 1
[[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> 2
[[1,2,3,4,5,9],[6,7,8]]
=> ? = 1
[[1,2,3,4,5,9],[6,7],[8]]
=> ? = 1
[[1,2,3,4,5,9],[6],[7],[8]]
=> ? = 1
[[1,2,3,4,9],[5,6,7],[8]]
=> ? = 1
[[1,2,3,4,9],[5,6],[7],[8]]
=> ? = 1
[[1,2,3,4,9],[5],[6],[7],[8]]
=> ? = 1
[[1,2,3,7],[4,5,6,9],[8]]
=> ? = 1
[[1,2,3,9],[4,5,6],[7,8]]
=> ? = 1
[[1,2,3,9],[4,5,6],[7],[8]]
=> ? = 1
[[1,2,3,9],[4,5],[6,7],[8]]
=> ? = 1
[[1,2,3,9],[4,5],[6],[7],[8]]
=> ? = 1
[[1,2,3,9],[4],[5],[6],[7],[8]]
=> ? = 1
[[1,2,5],[3,4,8],[6,7,9]]
=> ? = 1
[[1,2,5],[3,4,9],[6,7],[8]]
=> ? = 1
[[1,2,5],[3,4,9],[6],[7],[8]]
=> ? = 1
[[1,2,9],[3,4],[5,6],[7],[8]]
=> ? = 1
[[1,2,9],[3,4],[5],[6],[7],[8]]
=> ? = 1
[[1,3],[2,5],[4,9],[6],[7],[8]]
=> ? = 2
[[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 2
[[1,2,3,4,5,6,7,10],[8,9]]
=> ? = 1
[[1,2,3,4,5,6,7,10],[8],[9]]
=> ? = 1
[[1,2,3,4,5,6,10],[7,8],[9]]
=> ? = 1
[[1,2,3,4,5,6,10],[7],[8],[9]]
=> ? = 1
[[1,2,3,4,5,10],[6,7],[8],[9]]
=> ? = 1
[[1,2,3,4,5,10],[6],[7],[8],[9]]
=> ? = 1
[[1,2,3,4,10],[5,6],[7],[8],[9]]
=> ? = 1
[[1,2,3,4,10],[5],[6],[7],[8],[9]]
=> ? = 1
[[1,2,3,10],[4,5,6],[7],[8],[9]]
=> ? = 1
[[1,2,3,10],[4,5],[6,7],[8],[9]]
=> ? = 1
[[1,2,3,10],[4,5],[6],[7],[8],[9]]
=> ? = 1
[[1,2,3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 1
[[1,2,5],[3,4,10],[6,7],[8],[9]]
=> ? = 1
[[1,2,10],[3,4],[5,6],[7,8],[9]]
=> ? = 1
[[1,2,10],[3,4],[5],[6],[7],[8],[9]]
=> ? = 1
[[1,3],[2,10],[4],[5],[6],[7],[8],[9]]
=> ? = 2
[[1,2,4,7,8,9],[3,5],[6]]
=> ? = 1
[[1,4,6,7,8,9],[2],[3],[5]]
=> ? = 3
[[1,3,5,7,8],[2,4,6,9]]
=> ? = 2
[[1,2,5,7,9],[3,6,8],[4]]
=> ? = 1
[[1,4,6,8,9],[2,5],[3,7]]
=> ? = 3
[[1,3,5,8,9],[2,6],[4],[7]]
=> ? = 2
[[1,5,7,8,9],[2],[3],[4],[6]]
=> ? = 4
[[1,2,5,6],[3,7,8,9],[4]]
=> ? = 1
[[1,3,4,7],[2,6,8],[5,9]]
=> ? = 2
[[1,3,6,8],[2,7,9],[4],[5]]
=> ? = 2
[[1,2,6,9],[3,4],[5,7],[8]]
=> ? = 1
[[1,4,6,9],[2,7],[3],[5],[8]]
=> ? = 3
[[1,6,8,9],[2],[3],[4],[5],[7]]
=> ? = 5
[[1,2,4],[3,5,7],[6,8],[9]]
=> ? = 1
[[1,4,7],[2,8,9],[3],[5],[6]]
=> ? = 3
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1
[[1,2]]
=> [2] => [1,1] => 1
[[1],[2]]
=> [1,1] => [2] => 2
[[1,2,3]]
=> [3] => [1,1,1] => 1
[[1,3],[2]]
=> [1,2] => [1,2] => 2
[[1,2],[3]]
=> [2,1] => [2,1] => 1
[[1],[2],[3]]
=> [1,1,1] => [3] => 3
[[1,2,3,4]]
=> [4] => [1,1,1,1] => 1
[[1,3,4],[2]]
=> [1,3] => [1,1,2] => 2
[[1,2,4],[3]]
=> [2,2] => [1,2,1] => 1
[[1,2,3],[4]]
=> [3,1] => [2,1,1] => 1
[[1,3],[2,4]]
=> [1,2,1] => [2,2] => 2
[[1,2],[3,4]]
=> [2,2] => [1,2,1] => 1
[[1,4],[2],[3]]
=> [1,1,2] => [1,3] => 3
[[1,3],[2],[4]]
=> [1,2,1] => [2,2] => 2
[[1,2],[3],[4]]
=> [2,1,1] => [3,1] => 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => 1
[[1,3,4,5],[2]]
=> [1,4] => [1,1,1,2] => 2
[[1,2,4,5],[3]]
=> [2,3] => [1,1,2,1] => 1
[[1,2,3,5],[4]]
=> [3,2] => [1,2,1,1] => 1
[[1,2,3,4],[5]]
=> [4,1] => [2,1,1,1] => 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => 2
[[1,2,5],[3,4]]
=> [2,3] => [1,1,2,1] => 1
[[1,3,4],[2,5]]
=> [1,3,1] => [2,1,2] => 2
[[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => 1
[[1,2,3],[4,5]]
=> [3,2] => [1,2,1,1] => 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,1,3] => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,2,2] => 2
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,3,1] => 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [2,1,2] => 2
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [3,1,1] => 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [2,3] => 3
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => 2
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,3,1] => 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [3,2] => 2
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,4] => 4
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [2,3] => 3
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [3,2] => 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [4,1] => 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1] => 1
[[1,3,4,5,6],[2]]
=> [1,5] => [1,1,1,1,2] => 2
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,1,2,1] => 1
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,2,1,1] => 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,2,1,1,1] => 1
[[1,2,3,4,5],[6]]
=> [5,1] => [2,1,1,1,1] => 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,1,2,2] => 2
[[1,3,4,5,6,7,8],[2]]
=> [1,7] => [1,1,1,1,1,1,2] => ? = 2
[[1,2,3,4,5,6,7],[8]]
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
[[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => [1,1,1,1,1,3] => ? = 3
[[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [3,1,1,1,1,1] => ? = 1
[[1,4,6,7,8],[2,5],[3]]
=> [1,1,2,4] => [1,1,1,2,3] => ? = 3
[[1,4,5,7,8],[2,6],[3]]
=> [1,1,3,3] => [1,1,2,1,3] => ? = 3
[[1,4,5,6,8],[2,7],[3]]
=> [1,1,4,2] => [1,2,1,1,3] => ? = 3
[[1,4,5,6,7],[2,8],[3]]
=> [1,1,5,1] => [2,1,1,1,3] => ? = 3
[[1,3,4,5,6],[2,7],[8]]
=> [1,5,1,1] => [3,1,1,1,2] => ? = 2
[[1,2,4,5,6],[3,7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => ? = 1
[[1,2,3,5,6],[4,7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => ? = 1
[[1,2,3,4,6],[5,7],[8]]
=> [4,2,1,1] => [3,2,1,1,1] => ? = 1
[[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => [1,1,1,1,4] => ? = 4
[[1,4,6,7,8],[2],[3],[5]]
=> [1,1,2,4] => [1,1,1,2,3] => ? = 3
[[1,2,6,7,8],[3],[4],[5]]
=> [2,1,1,4] => [1,1,1,4,1] => ? = 1
[[1,4,5,7,8],[2],[3],[6]]
=> [1,1,3,3] => [1,1,2,1,3] => ? = 3
[[1,2,3,7,8],[4],[5],[6]]
=> [3,1,1,3] => [1,1,4,1,1] => ? = 1
[[1,4,5,6,8],[2],[3],[7]]
=> [1,1,4,2] => [1,2,1,1,3] => ? = 3
[[1,2,3,4,8],[5],[6],[7]]
=> [4,1,1,2] => [1,4,1,1,1] => ? = 1
[[1,4,5,6,7],[2],[3],[8]]
=> [1,1,5,1] => [2,1,1,1,3] => ? = 3
[[1,3,4,5,6],[2],[7],[8]]
=> [1,5,1,1] => [3,1,1,1,2] => ? = 2
[[1,2,4,5,6],[3],[7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => ? = 1
[[1,2,3,5,6],[4],[7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => ? = 1
[[1,2,3,4,6],[5],[7],[8]]
=> [4,2,1,1] => [3,2,1,1,1] => ? = 1
[[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [4,1,1,1,1] => ? = 1
[[1,4,5,8],[2,6,7],[3]]
=> [1,1,3,3] => [1,1,2,1,3] => ? = 3
[[1,4,5,6],[2,7,8],[3]]
=> [1,1,4,2] => [1,2,1,1,3] => ? = 3
[[1,2,5,6],[3,4,7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => ? = 1
[[1,2,3,6],[4,5,7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => ? = 1
[[1,4,7,8],[2,5],[3,6]]
=> [1,1,2,4] => [1,1,1,2,3] => ? = 3
[[1,4,5,8],[2,6],[3,7]]
=> [1,1,3,3] => [1,1,2,1,3] => ? = 3
[[1,4,5,6],[2,7],[3,8]]
=> [1,1,4,2] => [1,2,1,1,3] => ? = 3
[[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 4
[[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => [1,1,1,2,3] => ? = 3
[[1,2,7,8],[3,6],[4],[5]]
=> [2,1,1,4] => [1,1,1,4,1] => ? = 1
[[1,3,7,8],[2,4],[5],[6]]
=> [1,2,1,1,3] => [1,1,4,2] => ? = 2
[[1,4,5,8],[2,7],[3],[6]]
=> [1,1,3,3] => [1,1,2,1,3] => ? = 3
[[1,2,3,8],[4,7],[5],[6]]
=> [3,1,1,3] => [1,1,4,1,1] => ? = 1
[[1,5,6,7],[2,8],[3],[4]]
=> [1,1,1,4,1] => [2,1,1,4] => ? = 4
[[1,2,3,7],[4,8],[5],[6]]
=> [3,1,1,2,1] => [2,4,1,1] => ? = 1
[[1,4,5,6],[2,8],[3],[7]]
=> [1,1,4,2] => [1,2,1,1,3] => ? = 3
[[1,2,3,4],[5,8],[6],[7]]
=> [4,1,1,2] => [1,4,1,1,1] => ? = 1
[[1,2,5,6],[3,4],[7],[8]]
=> [2,4,1,1] => [3,1,1,2,1] => ? = 1
[[1,2,3,6],[4,5],[7],[8]]
=> [3,3,1,1] => [3,1,2,1,1] => ? = 1
[[1,3,4,5],[2,6],[7],[8]]
=> [1,4,1,1,1] => [4,1,1,2] => ? = 2
[[1,2,3,5],[4,6],[7],[8]]
=> [3,2,1,1,1] => [4,2,1,1] => ? = 1
[[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => [3,2,1,1,1] => ? = 1
[[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => [1,1,1,5] => ? = 5
[[1,5,7,8],[2],[3],[4],[6]]
=> [1,1,1,2,3] => [1,1,2,4] => ? = 4
[[1,3,7,8],[2],[4],[5],[6]]
=> [1,2,1,1,3] => [1,1,4,2] => ? = 2
Description
The last part of an integer composition.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[[1,2]]
=> [2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[[1],[2]]
=> [1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[[1,2,3]]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3],[2]]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3]]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,3,4]]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3,4],[2]]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,4],[3]]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3,4]]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2],[4]]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3,4,5],[2]]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,4,5],[3]]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[4,5]]
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 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]
=> 2 = 1 + 1
[[1,3,4,5,6],[2]]
=> [1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[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]
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4,6,7,8],[2,5],[3]]
=> [1,1,2,4] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,5,7,8],[2,6],[3]]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,5,6,8],[2,7],[3]]
=> [1,1,4,2] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[1,4,6,7,8],[2],[3],[5]]
=> [1,1,2,4] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,5,7,8],[2],[3],[6]]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,5,6,8],[2],[3],[7]]
=> [1,1,4,2] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,6,8],[2,5,7],[3]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5,8],[2,6,7],[3]]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,6,7],[2,5,8],[3]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,7],[2,6,8],[3]]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,6],[2,7,8],[3]]
=> [1,1,4,2] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,7,8],[2,5],[3,6]]
=> [1,1,2,4] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,6,8],[2,5],[3,7]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5,8],[2,6],[3,7]]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,6,7],[2,5],[3,8]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,7],[2,6],[3,8]]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,6],[2,7],[3,8]]
=> [1,1,4,2] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,5,7,8],[2,6],[3],[4]]
=> [1,1,1,2,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,7,8],[2,5],[3],[6]]
=> [1,1,2,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,5,6,8],[2,7],[3],[4]]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,4,6,8],[2,7],[3],[5]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5,8],[2,7],[3],[6]]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,6,8],[2,5],[3],[7]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5,8],[2,6],[3],[7]]
=> [1,1,3,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,5,6,7],[2,8],[3],[4]]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 + 1
[[1,4,6,7],[2,8],[3],[5]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,7],[2,8],[3],[6]]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,6],[2,8],[3],[7]]
=> [1,1,4,2] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,6,7],[2,5],[3],[8]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,7],[2,6],[3],[8]]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[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]
=> ? = 3 + 1
[[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[[1,5,7,8],[2],[3],[4],[6]]
=> [1,1,1,2,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[1,4,7,8],[2],[3],[5],[6]]
=> [1,1,2,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,5,6,8],[2],[3],[4],[7]]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,4,6,8],[2],[3],[5],[7]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5,8],[2],[3],[6],[7]]
=> [1,1,3,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,5,6,7],[2],[3],[4],[8]]
=> [1,1,1,4,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 + 1
[[1,4,6,7],[2],[3],[5],[8]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,5,7],[2],[3],[6],[8]]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[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]
=> ? = 3 + 1
[[1,4,7],[2,5,8],[3,6]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,6],[2,5,8],[3,7]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5],[2,6,8],[3,7]]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,6],[2,5,7],[3,8]]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5],[2,6,7],[3,8]]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,7],[2,6,8],[3],[5]]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,4,7],[2,5,8],[3],[6]]
=> [1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 89%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 89%
Values
[[1]]
=> [1] => [1] => [1,0]
=> ? = 1
[[1,2]]
=> [2] => [2] => [1,1,0,0]
=> 1
[[1],[2]]
=> [1,1] => [1,1] => [1,0,1,0]
=> 2
[[1,2,3]]
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
[[1,3],[2]]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[[1],[2],[3]]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[[1,2,3,4]]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[[1,3,4],[2]]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4]]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,3],[2,4]]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5]]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3,4,5],[2]]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,4,5],[3]]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,2,3,5],[4]]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,4],[5]]
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,3,4],[2,5]]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3,5]]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4,5]]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3],[4]]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,5],[4]]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,3,4,5,6]]
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,2,4,5,6],[3]]
=> [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,4,6],[5]]
=> [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5,6],[3,4]]
=> [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4,5,6,7,8]]
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[[1,3,4,5,6,7,8],[2]]
=> [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,6,7,8],[3]]
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,5,6,7,8],[4]]
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,2,3,4,6,7,8],[5]]
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,2,3,4,5,7,8],[6]]
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,3,5,6,7,8],[2,4]]
=> [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,5,6,7,8],[3,4]]
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[1,3,4,6,7,8],[2,5]]
=> [1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,2,4,6,7,8],[3,5]]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,6,7,8],[4,5]]
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,4,5,7,8],[2,6]]
=> [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,7,8],[3,6]]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,5,7,8],[4,6]]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,2,3,4,7,8],[5,6]]
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,2,3,4,5,8],[6,7]]
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,5,6,7,8],[2],[4]]
=> [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,5,6,7,8],[3],[4]]
=> [2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,3,4,6,7,8],[2],[5]]
=> [1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,2,4,6,7,8],[3],[5]]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,6,7,8],[4],[5]]
=> [3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,4,5,7,8],[2],[6]]
=> [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,7,8],[3],[6]]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,5,7,8],[4],[6]]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,2,3,4,7,8],[5],[6]]
=> [4,1,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,3,5,7,8],[2,4,6]]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[1,3,4,7,8],[2,5,6]]
=> [1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,2,4,7,8],[3,5,6]]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,7,8],[4,5,6]]
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,4,5,8],[2,6,7]]
=> [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,5,8],[4,6,7]]
=> [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,2,3,4,8],[5,6,7]]
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,2,3,4,5],[6,7,8]]
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,4,6,7,8],[2,5],[3]]
=> [1,1,2,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,6,7,8],[3,5],[4]]
=> [2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,3,6,7,8],[2,4],[5]]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,6,7,8],[3,4],[5]]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,4,5,7,8],[2,6],[3]]
=> [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,5,7,8],[2,6],[4]]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,5,7,8],[3,6],[4]]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,3,4,7,8],[2,6],[5]]
=> [1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,2,4,7,8],[3,6],[5]]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,7,8],[4,6],[5]]
=> [3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,5,7,8],[2,4],[6]]
=> [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[1,2,5,7,8],[3,4],[6]]
=> [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000054
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1] => 1
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,3,2] => 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 3
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [1,4,3,2] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => [1,4,3,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => [1,4,3,2] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,5,4,3,2] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => [2,1,5,4,3] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,5,4,3,2] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,5,4,3,2] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,5,4,3,2] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,5,4,3,2] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,5,4,3,2] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,6,5,4,3] => 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,6,5,4,3,2] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,6,5,4,3,2] => 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,6,5,4,3,2] => 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,6,5,4,3,2] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => [2,1,6,5,4,3] => 2
[[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [3,2,1,4,6,5,7] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,5,6],[2,7],[3]]
=> [3,2,7,1,4,5,6] => [3,2,1,4,5,7,6] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => [2,7,1,6,5,4,3] => ? = 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ? = 4
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => [3,7,2,1,6,5,4] => ? = 3
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => [2,7,6,1,5,4,3] => ? = 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [3,2,6,1,4,5,7] => [3,2,7,1,6,5,4] => ? = 3
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => [2,7,6,1,5,4,3] => ? = 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [3,2,1,7,4,5,6] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [3,2,1,4,6,7,5] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [2,4,1,3,6,7,5] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,2,1,6,5,4,7] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [2,4,1,6,5,3,7] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,2,1,5,7,4,6] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,2,1,4,7,6,5] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,1,3,7,6,5] => [2,7,1,6,5,4,3] => ? = 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => ? = 4
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,5,2,1,6,4,7] => [3,7,2,1,6,5,4] => ? = 3
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [2,5,4,1,6,3,7] => [2,7,6,1,5,4,3] => ? = 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [3,2,6,1,5,4,7] => [3,2,7,1,6,5,4] => ? = 3
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [2,4,6,1,5,3,7] => [2,7,6,1,5,4,3] => ? = 2
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => ? = 4
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,2,1,4,7,6] => [3,7,2,1,6,5,4] => ? = 3
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,4,1,3,7,6] => [2,7,6,1,5,4,3] => ? = 2
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [3,2,6,1,4,7,5] => [3,2,7,1,6,5,4] => ? = 3
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => [2,7,6,1,5,4,3] => ? = 2
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => [2,7,1,6,5,4,3] => ? = 2
[[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => [3,2,1,7,4,6,5] => [3,2,1,7,6,5,4] => ? = 3
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => [2,7,1,6,5,4,3] => ? = 2
[[1,5,7],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7] => [4,6,3,2,1,5,7] => [4,7,3,2,1,6,5] => ? = 4
[[1,4,7],[2],[3],[5],[6]]
=> [6,5,3,2,1,4,7] => [3,6,5,2,1,4,7] => [3,7,6,2,1,5,4] => ? = 3
[[1,3,7],[2],[4],[5],[6]]
=> [6,5,4,2,1,3,7] => [2,6,5,4,1,3,7] => [2,7,6,5,1,4,3] => ? = 2
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [4,3,7,2,1,5,6] => [4,3,7,2,1,6,5] => ? = 4
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => [3,7,6,2,1,5,4] => ? = 3
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => [2,7,6,5,1,4,3] => ? = 2
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [3,2,7,6,1,4,5] => [3,2,7,6,1,5,4] => ? = 3
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => [2,7,6,5,1,4,3] => ? = 2
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 4
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [3,5,2,1,7,6,4] => [3,7,2,1,6,5,4] => ? = 3
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => [2,7,6,1,5,4,3] => ? = 2
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).
Matching statistic: St000617
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,0]
=> 1
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 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]
=> 2
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[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]
=> 1
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[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]
=> 3
[[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]
=> 2
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[[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]
=> 1
[[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]
=> 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[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]
=> 2
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[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]
=> 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[[1,2,5,6,7],[3],[4]]
=> [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[[1,2,3,6,7],[4],[5]]
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[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]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,2,6,7],[3,5],[4]]
=> [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 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]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 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]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[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]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3,7],[4]]
=> [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,3,4,6],[2,7],[5]]
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,6],[4,7],[5]]
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,3,4,5],[2,6],[7]]
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,5],[3,6],[7]]
=> [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[[1,2,3,5],[4,6],[7]]
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[[1,4,6,7],[2],[3],[5]]
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 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]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[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]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,2,3,7],[4],[5],[6]]
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[1,2,5,6],[3],[4],[7]]
=> [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[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]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,2,3,6],[4],[5],[7]]
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[[1,3,4,5],[2],[6],[7]]
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[[1,2,4,5],[3],[6],[7]]
=> [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[[1,2,3,5],[4],[6],[7]]
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,6],[2,5,7],[3]]
=> [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[[1,2,6],[3,5,7],[4]]
=> [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,3,6],[2,4,7],[5]]
=> [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,2,3],[4,6,7],[5]]
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[1,3,4],[2,5,7],[6]]
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[1,2,4],[3,5,7],[6]]
=> [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,3,5],[2,4,6],[7]]
=> [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[1,2,5],[3,4,6],[7]]
=> [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[[1,3,4],[2,5,6],[7]]
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,4,7],[2,5],[3,6]]
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,3,7],[2,4],[5,6]]
=> [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[1,4,6],[2,5],[3,7]]
=> [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[[1,2,6],[3,5],[4,7]]
=> [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[[1,3,6],[2,4],[5,7]]
=> [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[1,3,4],[2,6],[5,7]]
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[[1,2,3],[4,6],[5,7]]
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[[1,3,4],[2,5],[6,7]]
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[1,2,4],[3,5],[6,7]]
=> [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[1,3,7],[2,6],[4],[5]]
=> [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[1,2,7],[3,4],[5],[6]]
=> [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St001050
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 78%
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 78%
Values
[[1]]
=> {{1}}
=> {{1}}
=> {{1}}
=> 1
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 1
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 2
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,3},{2,4}}
=> 2
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> 2
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,3},{2,4,5}}
=> 1
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> {{1,2,5},{3,4}}
=> 2
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,2,4,5},{3}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 2
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> 3
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> 2
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 1
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,4},{3,5}}
=> 2
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 3
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 2
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 4
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 3
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 2
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 1
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 5
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 1
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,3,4,5,6},{2}}
=> {{1,3,5},{2,4,6}}
=> 2
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,4,5,6},{3}}
=> {{1,3,5,6},{2,4}}
=> 1
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3,5,6},{4}}
=> {{1,3},{2,4,5,6}}
=> 1
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4,6},{5}}
=> {{1,3,4,5,6},{2}}
=> 1
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,4},{2,3,5,6}}
=> {{1,2,3,6},{4,5}}
=> 2
[[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> {{1,3,4,5,6,7,8},{2}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 2
[[1,2,4,5,6,7,8],[3]]
=> {{1,2,4,5,6,7,8},{3}}
=> {{1,2,4,5,6,7,8},{3}}
=> {{1,3,5,7,8},{2,4,6}}
=> ? = 1
[[1,2,3,5,6,7,8],[4]]
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,3,5},{2,4,6,7,8}}
=> ? = 1
[[1,3,4,6,7,8],[2,5]]
=> {{1,3,4,6,7,8},{2,5}}
=> {{1,5},{2,3,4,6,7,8}}
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[[1,3,4,5,7,8],[2,6]]
=> {{1,3,4,5,7,8},{2,6}}
=> {{1,6},{2,3,4,5,7,8}}
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[[1,2,4,5,7,8],[3,6]]
=> {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6},{3,4,5,7,8}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1
[[1,3,4,5,6,8],[2,7]]
=> {{1,3,4,5,6,8},{2,7}}
=> {{1,7},{2,3,4,5,6,8}}
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 2
[[1,2,4,5,6,8],[3,7]]
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,2,7},{3,4,5,6,8}}
=> {{1,2,7,8},{3,4,5,6}}
=> ? = 1
[[1,2,3,5,6,8],[4,7]]
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7},{4,5,6,8}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 1
[[1,2,4,5,6,7],[3,8]]
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 1
[[1,2,3,5,6,7],[4,8]]
=> {{1,2,3,5,6,7},{4,8}}
=> {{1,2,3,8},{4,5,6,7}}
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 1
[[1,2,3,4,6,7],[5,8]]
=> {{1,2,3,4,6,7},{5,8}}
=> {{1,2,3,4,8},{5,6,7}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 1
[[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> {{1,4,5,6,7,8},{2},{3}}
=> {{1,4,7},{2,5,8},{3,6}}
=> ? = 3
[[1,3,5,6,7,8],[2],[4]]
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1,4,7},{2,5},{3,6,8}}
=> ? = 2
[[1,2,5,6,7,8],[3],[4]]
=> {{1,2,5,6,7,8},{3},{4}}
=> {{1,2,5,6,7,8},{3},{4}}
=> {{1,4,7,8},{2,5},{3,6}}
=> ? = 1
[[1,3,4,6,7,8],[2],[5]]
=> {{1,3,4,6,7,8},{2},{5}}
=> {{1,3,4,6,7,8},{2},{5}}
=> {{1,4},{2,5,7},{3,6,8}}
=> ? = 2
[[1,2,4,6,7,8],[3],[5]]
=> {{1,2,4,6,7,8},{3},{5}}
=> {{1,2,4,6,7,8},{3},{5}}
=> {{1,4},{2,5,7,8},{3,6}}
=> ? = 1
[[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> {{1,2,3,6,7,8},{4},{5}}
=> {{1,4},{2,5},{3,6,7,8}}
=> ? = 1
[[1,3,4,5,7,8],[2],[6]]
=> {{1,3,4,5,7,8},{2},{6}}
=> {{1,3,4,5,7,8},{2},{6}}
=> {{1,4,6,8},{2,5,7},{3}}
=> ? = 2
[[1,2,4,5,7,8],[3],[6]]
=> {{1,2,4,5,7,8},{3},{6}}
=> {{1,2,4,5,7,8},{3},{6}}
=> {{1,4,6},{2,5,7,8},{3}}
=> ? = 1
[[1,2,3,5,7,8],[4],[6]]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,4,6,7,8},{2,5},{3}}
=> ? = 1
[[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,4},{2,5,6,7,8},{3}}
=> ? = 1
[[1,3,4,5,6,8],[2],[7]]
=> {{1,3,4,5,6,8},{2},{7}}
=> {{1,3,4,5,6,8},{2},{7}}
=> {{1,4,6,8},{2},{3,5,7}}
=> ? = 2
[[1,2,4,5,6,8],[3],[7]]
=> {{1,2,4,5,6,8},{3},{7}}
=> {{1,2,4,5,6,8},{3},{7}}
=> {{1,4,6},{2},{3,5,7,8}}
=> ? = 1
[[1,2,3,5,6,8],[4],[7]]
=> {{1,2,3,5,6,8},{4},{7}}
=> {{1,2,3,5,6,8},{4},{7}}
=> {{1,4,6,7,8},{2},{3,5}}
=> ? = 1
[[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> {{1,2,3,4,6,8},{5},{7}}
=> {{1,4},{2},{3,5,6,7,8}}
=> ? = 1
[[1,3,4,5,6,7],[2],[8]]
=> {{1,3,4,5,6,7},{2},{8}}
=> {{1,3,4,5,6,7},{2},{8}}
=> {{1},{2,4,6,8},{3,5,7}}
=> ? = 2
[[1,2,4,5,6,7],[3],[8]]
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1},{2,4,6},{3,5,7,8}}
=> ? = 1
[[1,2,3,5,6,7],[4],[8]]
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1},{2,4,6,7,8},{3,5}}
=> ? = 1
[[1,2,3,4,6,7],[5],[8]]
=> {{1,2,3,4,6,7},{5},{8}}
=> {{1,2,3,4,6,7},{5},{8}}
=> {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5,7,8},{2,3,6}}
=> {{1,3,4,6,7},{2,5,8}}
=> ? = 2
[[1,2,5,7,8],[3,4,6]]
=> {{1,2,5,7,8},{3,4,6}}
=> {{1,2,4,5,7,8},{3,6}}
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 1
[[1,3,4,7,8],[2,5,6]]
=> {{1,3,4,7,8},{2,5,6}}
=> {{1,5,7,8},{2,3,4,6}}
=> {{1,3,4,8},{2,5,6,7}}
=> ? = 2
[[1,2,4,7,8],[3,5,6]]
=> {{1,2,4,7,8},{3,5,6}}
=> {{1,2,5,7,8},{3,4,6}}
=> {{1,3,4,7,8},{2,5,6}}
=> ? = 1
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> {{1,4,5,6,8},{2,3,7}}
=> {{1,4,6,7},{2,3,5,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,4,6},{2,3,5,7,8}}
=> ? = 1
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> {{1,5,6,8},{2,3,4,7}}
=> {{1,4,8},{2,3,5,6,7}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,4,7,8},{2,3,5,6}}
=> ? = 1
[[1,2,3,6,8],[4,5,7]]
=> {{1,2,3,6,8},{4,5,7}}
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,4,6,7,8},{2,3,5}}
=> ? = 1
[[1,3,4,5,8],[2,6,7]]
=> {{1,3,4,5,8},{2,6,7}}
=> {{1,6,8},{2,3,4,5,7}}
=> {{1,4,5,6,7},{2,3,8}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> {{1,2,4,5,8},{3,6,7}}
=> {{1,2,6,8},{3,4,5,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> ? = 1
[[1,2,3,5,8],[4,6,7]]
=> {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6,8},{4,5,7}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 1
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> {{1,4,5,6,7},{2,3,8}}
=> {{1,2,4,6,7},{3,5,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 1
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> {{1,5,6,7},{2,3,4,8}}
=> {{1,2,4,8},{3,5,6,7}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,7,8},{3,5,6}}
=> ? = 1
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> {{1,2,4,5,6},{3,7,8}}
=> ? = 1
[[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 1
[[1,2,3,4,7],[5,6,8]]
=> {{1,2,3,4,7},{5,6,8}}
=> {{1,2,3,4,6,7},{5,8}}
=> {{1,2,4},{3,5,6,7,8}}
=> ? = 1
[[1,3,4,5,6],[2,7,8]]
=> {{1,3,4,5,6},{2,7,8}}
=> ?
=> ?
=> ? = 2
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000759The smallest missing part in an integer 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. St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. 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. St000068The number of minimal elements in a poset. St000501The size of the first part in the decomposition of a permutation. St000654The first descent of a permutation. St000542The number of left-to-right-minima 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. St000990The first ascent of a permutation. St000989The number of final rises of a permutation. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. 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. St001481The minimal height of a peak of a Dyck path. 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.
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