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Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000507
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St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,2]]
=> 2
[[1],[2]]
=> 1
[[1,2,3]]
=> 3
[[1,3],[2]]
=> 2
[[1,2],[3]]
=> 2
[[1],[2],[3]]
=> 1
[[1,2,3,4]]
=> 4
[[1,3,4],[2]]
=> 3
[[1,2,4],[3]]
=> 3
[[1,2,3],[4]]
=> 3
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 3
[[1,4],[2],[3]]
=> 2
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 1
[[1,2,3,4,5]]
=> 5
[[1,3,4,5],[2]]
=> 4
[[1,2,4,5],[3]]
=> 4
[[1,2,3,5],[4]]
=> 4
[[1,2,3,4],[5]]
=> 4
[[1,3,5],[2,4]]
=> 3
[[1,2,5],[3,4]]
=> 4
[[1,3,4],[2,5]]
=> 3
[[1,2,4],[3,5]]
=> 3
[[1,2,3],[4,5]]
=> 4
[[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> 3
[[1,2,5],[3],[4]]
=> 3
[[1,3,4],[2],[5]]
=> 3
[[1,2,4],[3],[5]]
=> 3
[[1,2,3],[4],[5]]
=> 3
[[1,4],[2,5],[3]]
=> 2
[[1,3],[2,5],[4]]
=> 3
[[1,2],[3,5],[4]]
=> 3
[[1,3],[2,4],[5]]
=> 2
[[1,2],[3,4],[5]]
=> 3
[[1,5],[2],[3],[4]]
=> 2
[[1,4],[2],[3],[5]]
=> 2
[[1,3],[2],[4],[5]]
=> 2
[[1,2],[3],[4],[5]]
=> 2
[[1],[2],[3],[4],[5]]
=> 1
[[1,2,3,4,5,6]]
=> 6
[[1,3,4,5,6],[2]]
=> 5
[[1,2,4,5,6],[3]]
=> 5
[[1,2,3,5,6],[4]]
=> 5
[[1,2,3,4,6],[5]]
=> 5
[[1,2,3,4,5],[6]]
=> 5
[[1,3,5,6],[2,4]]
=> 4
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000288
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(load all 14 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 55% ●values known / values provided: 72%●distinct values known / distinct values provided: 55%
Mp00224: Binary words —runsort⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 55% ●values known / values provided: 72%●distinct values known / distinct values provided: 55%
Values
[[1]]
=> => => => ? = 1 - 1
[[1,2]]
=> 0 => 0 => 1 => 1 = 2 - 1
[[1],[2]]
=> 1 => 1 => 0 => 0 = 1 - 1
[[1,2,3]]
=> 00 => 00 => 11 => 2 = 3 - 1
[[1,3],[2]]
=> 10 => 01 => 10 => 1 = 2 - 1
[[1,2],[3]]
=> 01 => 01 => 10 => 1 = 2 - 1
[[1],[2],[3]]
=> 11 => 11 => 00 => 0 = 1 - 1
[[1,2,3,4]]
=> 000 => 000 => 111 => 3 = 4 - 1
[[1,3,4],[2]]
=> 100 => 001 => 110 => 2 = 3 - 1
[[1,2,4],[3]]
=> 010 => 001 => 110 => 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => 001 => 110 => 2 = 3 - 1
[[1,3],[2,4]]
=> 101 => 011 => 100 => 1 = 2 - 1
[[1,2],[3,4]]
=> 010 => 001 => 110 => 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => 011 => 100 => 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => 011 => 100 => 1 = 2 - 1
[[1,2],[3],[4]]
=> 011 => 011 => 100 => 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => 111 => 000 => 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => 0000 => 1111 => 4 = 5 - 1
[[1,3,4,5],[2]]
=> 1000 => 0001 => 1110 => 3 = 4 - 1
[[1,2,4,5],[3]]
=> 0100 => 0001 => 1110 => 3 = 4 - 1
[[1,2,3,5],[4]]
=> 0010 => 0001 => 1110 => 3 = 4 - 1
[[1,2,3,4],[5]]
=> 0001 => 0001 => 1110 => 3 = 4 - 1
[[1,3,5],[2,4]]
=> 1010 => 0011 => 1100 => 2 = 3 - 1
[[1,2,5],[3,4]]
=> 0100 => 0001 => 1110 => 3 = 4 - 1
[[1,3,4],[2,5]]
=> 1001 => 0011 => 1100 => 2 = 3 - 1
[[1,2,4],[3,5]]
=> 0101 => 0101 => 1010 => 2 = 3 - 1
[[1,2,3],[4,5]]
=> 0010 => 0001 => 1110 => 3 = 4 - 1
[[1,4,5],[2],[3]]
=> 1100 => 0011 => 1100 => 2 = 3 - 1
[[1,3,5],[2],[4]]
=> 1010 => 0011 => 1100 => 2 = 3 - 1
[[1,2,5],[3],[4]]
=> 0110 => 0011 => 1100 => 2 = 3 - 1
[[1,3,4],[2],[5]]
=> 1001 => 0011 => 1100 => 2 = 3 - 1
[[1,2,4],[3],[5]]
=> 0101 => 0101 => 1010 => 2 = 3 - 1
[[1,2,3],[4],[5]]
=> 0011 => 0011 => 1100 => 2 = 3 - 1
[[1,4],[2,5],[3]]
=> 1101 => 0111 => 1000 => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> 1010 => 0011 => 1100 => 2 = 3 - 1
[[1,2],[3,5],[4]]
=> 0110 => 0011 => 1100 => 2 = 3 - 1
[[1,3],[2,4],[5]]
=> 1011 => 0111 => 1000 => 1 = 2 - 1
[[1,2],[3,4],[5]]
=> 0101 => 0101 => 1010 => 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> 1110 => 0111 => 1000 => 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => 0111 => 1000 => 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> 1011 => 0111 => 1000 => 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> 0111 => 0111 => 1000 => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => 1111 => 0000 => 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => 00000 => 11111 => 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> 10000 => 00001 => 11110 => 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> 01000 => 00001 => 11110 => 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> 00100 => 00001 => 11110 => 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> 00010 => 00001 => 11110 => 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> 00001 => 00001 => 11110 => 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> 10100 => 00011 => 11100 => 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> 01000 => 00001 => 11110 => 4 = 5 - 1
[[1,3,5,6,9],[2,4,7,8,10]]
=> 101001001 => 001001011 => 110110100 => ? = 6 - 1
[[1,3,4,7,9],[2,5,6,8,10]]
=> 100100101 => 001001011 => 110110100 => ? = 6 - 1
[[1,3,4,7,8],[2,5,6,9,10]]
=> 100100010 => 000010011 => 111101100 => ? = 7 - 1
[[1,3,4,6,9],[2,5,7,8,10]]
=> 100101001 => 001001011 => 110110100 => ? = 6 - 1
[[1,3,4,6,7],[2,5,8,9,10]]
=> 100100100 => 000010011 => 111101100 => ? = 7 - 1
[[1,3,4,5,9],[2,6,7,8,10]]
=> 100010001 => 000100011 => 111011100 => ? = 7 - 1
[[1,3,4,5,8],[2,6,7,9,10]]
=> 100010010 => 000010011 => 111101100 => ? = 7 - 1
[[1,2,5,7,8],[3,4,6,9,10]]
=> 010010010 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,5,6,9],[3,4,7,8,10]]
=> 010001001 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,4,5,9],[3,6,7,8,10]]
=> 010010001 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,4,5,8],[3,6,7,9,10]]
=> 010010010 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,3,7,9],[4,5,6,8,10]]
=> 001000101 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,3,7,8],[4,5,6,9,10]]
=> 001000010 => 000001001 => 111110110 => ? = 8 - 1
[[1,2,3,6,8],[4,5,7,9,10]]
=> 001001010 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,3,6,7],[4,5,8,9,10]]
=> 001000100 => 000001001 => 111110110 => ? = 8 - 1
[[1,2,3,5,9],[4,6,7,8,10]]
=> 001010001 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,3,5,8],[4,6,7,9,10]]
=> 001010010 => 000100101 => 111011010 => ? = 7 - 1
[[1,2,3,5,6],[4,7,8,9,10]]
=> 001001000 => 000001001 => 111110110 => ? = 8 - 1
[[1,2,3,4,7],[5,6,8,9,10]]
=> 000100100 => 000001001 => 111110110 => ? = 8 - 1
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> 10101010101 => 01010101011 => 10101010100 => ? = 6 - 1
[[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> 10101010010 => 00010101011 => 11101010100 => ? = 7 - 1
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> 10101001001 => 00100101011 => 11011010100 => ? = 7 - 1
[[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> 10101001010 => 00010101011 => 11101010100 => ? = 7 - 1
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> 10101000100 => 00000101011 => 11111010100 => ? = 8 - 1
[[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> 10100100101 => 00100101011 => 11011010100 => ? = 7 - 1
[[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> 10100100010 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> 10100101001 => 00100101011 => 11011010100 => ? = 7 - 1
[[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> 10100101010 => 00010101011 => 11101010100 => ? = 7 - 1
[[1,3,5,6,8,9],[2,4,7,10,11,12]]
=> 10100100100 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
=> 10100010001 => 00010001011 => ? => ? = 8 - 1
[[1,3,5,6,7,10],[2,4,8,9,11,12]]
=> 10100010010 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
=> 10100010100 => 00000101011 => 11111010100 => ? = 8 - 1
[[1,3,5,6,7,8],[2,4,9,10,11,12]]
=> 10100001000 => 00000001011 => 11111110100 => ? = 9 - 1
[[1,3,4,7,9,11],[2,5,6,8,10,12]]
=> 10010010101 => 00100101011 => 11011010100 => ? = 7 - 1
[[1,3,4,7,9,10],[2,5,6,8,11,12]]
=> 10010010010 => 00010010011 => ? => ? = 8 - 1
[[1,3,4,7,8,11],[2,5,6,9,10,12]]
=> 10010001001 => 00010010011 => ? => ? = 8 - 1
[[1,3,4,7,8,10],[2,5,6,9,11,12]]
=> 10010001010 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,4,7,8,9],[2,5,6,10,11,12]]
=> 10010000100 => 00000010011 => 11111101100 => ? = 9 - 1
[[1,3,4,6,9,11],[2,5,7,8,10,12]]
=> 10010100101 => 00100101011 => 11011010100 => ? = 7 - 1
[[1,3,4,6,9,10],[2,5,7,8,11,12]]
=> 10010100010 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,4,6,8,11],[2,5,7,9,10,12]]
=> 10010101001 => 00100101011 => 11011010100 => ? = 7 - 1
[[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> 10010101010 => 00010101011 => 11101010100 => ? = 7 - 1
[[1,3,4,6,8,9],[2,5,7,10,11,12]]
=> 10010100100 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,4,6,7,11],[2,5,8,9,10,12]]
=> 10010010001 => 00010010011 => ? => ? = 8 - 1
[[1,3,4,6,7,10],[2,5,8,9,11,12]]
=> 10010010010 => 00010010011 => ? => ? = 8 - 1
[[1,3,4,6,7,9],[2,5,8,10,11,12]]
=> 10010010100 => 00001001011 => 11110110100 => ? = 8 - 1
[[1,3,4,6,7,8],[2,5,9,10,11,12]]
=> 10010001000 => 00000010011 => 11111101100 => ? = 9 - 1
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
=> 10001000101 => 00010001011 => ? => ? = 8 - 1
[[1,3,4,5,9,10],[2,6,7,8,11,12]]
=> 10001000010 => 00000100011 => 11111011100 => ? = 9 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000010
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 65%
Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 65%
Values
[[1]]
=> [1] => [1] => [1]
=> 1
[[1,2]]
=> [1,2] => [1,2] => [1,1]
=> 2
[[1],[2]]
=> [2,1] => [2,1] => [2]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [2,1]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [2,1,1]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [2,1,1]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [2,2]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [3,1]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,1]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,1,1,1]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,1,1,1]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => [2,2,1]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [2,1,1,1]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => [2,2,1]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => [2,2,1]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [2,1,1,1]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [3,1,1]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,1,1]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,1,1]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [3,1,1]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [3,1,1]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => [3,2]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [3,1,1]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => [3,1,1]
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => [3,2]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => [3,1,1]
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [4,1]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [4,1]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [4,1]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,1,1,1]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => [2,1,1,1,1]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => [2,1,1,1,1]
=> 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => [2,1,1,1,1]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [2,1,1,1,1]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [2,2,1,1]
=> 4
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [3,1,4,5,6,2,7,8] => ?
=> ? = 6
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => [3,4,5,6,2,1,7,8] => ?
=> ? = 6
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [3,4,5,2,6,7,1,8] => ?
=> ? = 6
[[1,2,5,7,8],[3,4,6]]
=> [3,4,6,1,2,5,7,8] => [4,5,1,2,6,3,7,8] => ?
=> ? = 6
[[1,3,5,6,8],[2,4,7]]
=> [2,4,7,1,3,5,6,8] => [4,1,5,2,6,7,3,8] => ?
=> ? = 5
[[1,3,4,6,8],[2,5,7]]
=> [2,5,7,1,3,4,6,8] => [4,1,5,6,2,7,3,8] => ?
=> ? = 5
[[1,3,4,6,7],[2,5,8]]
=> [2,5,8,1,3,4,6,7] => [4,1,5,6,2,7,8,3] => ?
=> ? = 5
[[1,2,5,7,8],[3,6],[4]]
=> [4,3,6,1,2,5,7,8] => [4,5,2,1,6,3,7,8] => ?
=> ? = 5
[[1,3,4,7,8],[2,6],[5]]
=> [5,2,6,1,3,4,7,8] => [4,2,5,6,1,3,7,8] => ?
=> ? = 6
[[1,2,3,7,8],[4,6],[5]]
=> [5,4,6,1,2,3,7,8] => [4,5,6,2,1,3,7,8] => ?
=> ? = 6
[[1,3,5,7,8],[2,4],[6]]
=> [6,2,4,1,3,5,7,8] => [4,2,5,3,6,1,7,8] => ?
=> ? = 5
[[1,2,5,7,8],[3,4],[6]]
=> [6,3,4,1,2,5,7,8] => [4,5,2,3,6,1,7,8] => ?
=> ? = 6
[[1,3,4,7,8],[2,5],[6]]
=> [6,2,5,1,3,4,7,8] => [4,2,5,6,3,1,7,8] => ?
=> ? = 5
[[1,2,3,7,8],[4,5],[6]]
=> [6,4,5,1,2,3,7,8] => [4,5,6,2,3,1,7,8] => ?
=> ? = 6
[[1,3,5,6,8],[2,7],[4]]
=> [4,2,7,1,3,5,6,8] => [4,2,5,1,6,7,3,8] => ?
=> ? = 5
[[1,3,5,6,8],[2,4],[7]]
=> [7,2,4,1,3,5,6,8] => [4,2,5,3,6,7,1,8] => ?
=> ? = 5
[[1,3,4,6,8],[2,5],[7]]
=> [7,2,5,1,3,4,6,8] => [4,2,5,6,3,7,1,8] => ?
=> ? = 5
[[1,2,3,6,8],[4,5],[7]]
=> [7,4,5,1,2,3,6,8] => [4,5,6,2,3,7,1,8] => ?
=> ? = 6
[[1,3,4,5,8],[2,6],[7]]
=> [7,2,6,1,3,4,5,8] => [4,2,5,6,7,3,1,8] => ?
=> ? = 5
[[1,2,4,6,7],[3,8],[5]]
=> [5,3,8,1,2,4,6,7] => [4,5,2,6,1,7,8,3] => ?
=> ? = 5
[[1,2,3,5,6],[4,8],[7]]
=> [7,4,8,1,2,3,5,6] => [4,5,6,2,7,8,1,3] => ?
=> ? = 6
[[1,3,4,6,7],[2,5],[8]]
=> [8,2,5,1,3,4,6,7] => [4,2,5,6,3,7,8,1] => ?
=> ? = 5
[[1,2,4,5,6],[3,7],[8]]
=> [8,3,7,1,2,4,5,6] => [4,5,2,6,7,8,3,1] => ?
=> ? = 5
[[1,3,4,7,8],[2],[5],[6]]
=> [6,5,2,1,3,4,7,8] => [4,3,5,6,2,1,7,8] => ?
=> ? = 5
[[1,2,4,7,8],[3],[5],[6]]
=> [6,5,3,1,2,4,7,8] => [4,5,3,6,2,1,7,8] => ?
=> ? = 5
[[1,3,5,6,8],[2],[4],[7]]
=> [7,4,2,1,3,5,6,8] => [4,3,5,2,6,7,1,8] => ?
=> ? = 5
[[1,2,5,6,8],[3],[4],[7]]
=> [7,4,3,1,2,5,6,8] => [4,5,3,2,6,7,1,8] => ?
=> ? = 5
[[1,2,3,6,8],[4],[5],[7]]
=> [7,5,4,1,2,3,6,8] => [4,5,6,3,2,7,1,8] => ?
=> ? = 5
[[1,3,4,5,8],[2],[6],[7]]
=> [7,6,2,1,3,4,5,8] => [4,3,5,6,7,2,1,8] => ?
=> ? = 5
[[1,2,4,5,8],[3],[6],[7]]
=> [7,6,3,1,2,4,5,8] => [4,5,3,6,7,2,1,8] => ?
=> ? = 5
[[1,2,3,5,8],[4],[6],[7]]
=> [7,6,4,1,2,3,5,8] => [4,5,6,3,7,2,1,8] => ?
=> ? = 5
[[1,2,4,6,7],[3],[5],[8]]
=> [8,5,3,1,2,4,6,7] => [4,5,3,6,2,7,8,1] => ?
=> ? = 5
[[1,3,4,5,7],[2],[6],[8]]
=> [8,6,2,1,3,4,5,7] => [4,3,5,6,7,2,8,1] => ?
=> ? = 5
[[1,2,4,5,6],[3],[7],[8]]
=> [8,7,3,1,2,4,5,6] => [4,5,3,6,7,8,2,1] => ?
=> ? = 5
[[1,2,3,5,6],[4],[7],[8]]
=> [8,7,4,1,2,3,5,6] => [4,5,6,3,7,8,2,1] => ?
=> ? = 5
[[1,4,5,8],[2,6,7],[3]]
=> [3,2,6,7,1,4,5,8] => [5,2,1,6,7,3,4,8] => ?
=> ? = 5
[[1,2,5,8],[3,6,7],[4]]
=> [4,3,6,7,1,2,5,8] => [5,6,2,1,7,3,4,8] => ?
=> ? = 5
[[1,3,5,8],[2,4,7],[6]]
=> [6,2,4,7,1,3,5,8] => [5,2,6,3,7,1,4,8] => ?
=> ? = 5
[[1,2,5,8],[3,4,7],[6]]
=> [6,3,4,7,1,2,5,8] => [5,6,2,3,7,1,4,8] => ?
=> ? = 6
[[1,3,4,8],[2,5,7],[6]]
=> [6,2,5,7,1,3,4,8] => [5,2,6,7,3,1,4,8] => ?
=> ? = 5
[[1,3,4,8],[2,5,6],[7]]
=> [7,2,5,6,1,3,4,8] => [5,2,6,7,3,4,1,8] => ?
=> ? = 5
[[1,4,6,7],[2,5,8],[3]]
=> [3,2,5,8,1,4,6,7] => [5,2,1,6,3,7,8,4] => ?
=> ? = 4
[[1,2,6,7],[3,5,8],[4]]
=> [4,3,5,8,1,2,6,7] => [5,6,2,1,3,7,8,4] => ?
=> ? = 5
[[1,4,5,7],[2,6,8],[3]]
=> [3,2,6,8,1,4,5,7] => [5,2,1,6,7,3,8,4] => ?
=> ? = 4
[[1,3,4,7],[2,6,8],[5]]
=> [5,2,6,8,1,3,4,7] => [5,2,6,7,1,3,8,4] => ?
=> ? = 5
[[1,2,5,7],[3,4,8],[6]]
=> [6,3,4,8,1,2,5,7] => [5,6,2,3,7,1,8,4] => ?
=> ? = 5
[[1,3,4,7],[2,5,8],[6]]
=> [6,2,5,8,1,3,4,7] => [5,2,6,7,3,1,8,4] => ?
=> ? = 4
[[1,3,5,6],[2,7,8],[4]]
=> [4,2,7,8,1,3,5,6] => [5,2,6,1,7,8,3,4] => ?
=> ? = 5
[[1,3,4,6],[2,7,8],[5]]
=> [5,2,7,8,1,3,4,6] => [5,2,6,7,1,8,3,4] => ?
=> ? = 5
[[1,2,4,5],[3,7,8],[6]]
=> [6,3,7,8,1,2,4,5] => [5,6,2,7,8,1,3,4] => ?
=> ? = 6
Description
The length of the partition.
Matching statistic: St000676
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 40%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 40%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1
[[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] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[[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] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[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] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 4
[[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 7
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 7
[[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,6,7,8],[4,5]]
=> [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 7
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 6
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 6
[[1,2,3,4,5,8],[6,7]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 7
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
[[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 6
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => ?
=> ?
=> ? = 6
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> ? = 6
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => ?
=> ?
=> ? = 6
[[1,3,4,5,6,8],[2],[7]]
=> [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,2,3,4,5,8],[6],[7]]
=> [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 5
[[1,2,5,7,8],[3,4,6]]
=> [3,4,6,1,2,5,7,8] => ?
=> ?
=> ? = 6
[[1,3,4,7,8],[2,5,6]]
=> [2,5,6,1,3,4,7,8] => ?
=> ?
=> ? = 6
[[1,2,4,7,8],[3,5,6]]
=> [3,5,6,1,2,4,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 6
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001007
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 4
[[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 7
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 7
[[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 6
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 6
[[1,2,3,6,7,8],[4,5]]
=> [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 7
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 6
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 6
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 6
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 6
[[1,2,3,4,5,8],[6,7]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 7
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 6
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 6
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => ?
=> ?
=> ? = 6
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => ?
=> ?
=> ? = 6
[[1,3,4,5,6,8],[2],[7]]
=> [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[1,2,3,4,5,8],[6],[7]]
=> [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 6
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 5
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001068
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[[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] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[[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] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [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] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [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] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 7
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 7
[[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,3,6,7,8],[4,5]]
=> [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 7
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,8],[6,7]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 7
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 6
[[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => ?
=> ?
=> ? = 6
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 6
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => ?
=> ?
=> ? = 6
[[1,3,4,5,6,8],[2],[7]]
=> [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 6
[[1,2,3,4,5,8],[6],[7]]
=> [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 6
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 6
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6
[[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000024
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 2 - 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 1
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 - 1
[[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[1,2,3,6,7,8],[4,5]]
=> [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 6 - 1
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 6 - 1
[[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 6 - 1
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 7 - 1
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 6 - 1
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 6 - 1
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 6 - 1
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,8],[6,7]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 - 1
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => ?
=> ?
=> ? = 6 - 1
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => ?
=> ?
=> ? = 6 - 1
[[1,3,4,5,6,8],[2],[7]]
=> [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,8],[6],[7]]
=> [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 35%
Values
[[1]]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 7 - 1
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,3,6,7,8],[4,5]]
=> [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 - 1
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 7 - 1
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,8],[6,7]]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [[.,[.,.]],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7 - 1
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => ?
=> ?
=> ? = 6 - 1
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => ?
=> ?
=> ? = 6 - 1
[[1,3,4,5,6,8],[2],[7]]
=> [7,2,1,3,4,5,6,8] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[[1,2,3,4,5,8],[6],[7]]
=> [7,6,1,2,3,4,5,8] => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 6 - 1
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 6 - 1
[[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 - 1
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000632
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 40%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 40%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
[[1,2]]
=> [1,2] => [2,1] => ([],2)
=> 1 = 2 - 1
[[1],[2]]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => ([],3)
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 3 = 4 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2 = 3 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 2 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1 = 2 - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => ([(4,5)],6)
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,5,3,1,4,2] => ([(2,5),(3,4),(3,5)],6)
=> 3 = 4 - 1
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 5 - 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [7,5,3,1,6,4,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4 - 1
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [7,4,2,1,6,5,3] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 5 - 1
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 6 - 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [7,6,3,1,5,2,4] => ([(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ? = 5 - 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [7,5,3,1,6,2,4] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ? = 4 - 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [7,5,3,1,4,2,6] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 4 - 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 4 - 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 4 - 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [6,4,1,7,5,2,3] => ([(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 3 - 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [6,3,1,7,5,2,4] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5)],7)
=> ? = 4 - 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [6,3,1,7,4,2,5] => ([(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> ? = 3 - 1
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [5,3,1,7,6,2,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 4 - 1
[[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [5,2,1,7,6,3,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(4,3)],7)
=> ? = 4 - 1
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [4,3,1,7,6,2,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
=> ? = 5 - 1
[[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [4,2,1,7,6,3,5] => ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 5 - 1
[[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [3,2,1,7,6,4,5] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ? = 5 - 1
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [5,3,1,7,4,2,6] => ([(0,5),(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ? = 4 - 1
[[1,2,5],[3,4,7],[6]]
=> [6,3,4,7,1,2,5] => [5,2,1,7,4,3,6] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ? = 5 - 1
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [4,2,1,7,5,3,6] => ([(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7)
=> ? = 4 - 1
[[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [3,2,1,7,5,4,6] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ? = 5 - 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [5,3,1,6,4,2,7] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,2,4],[3,5,6],[7]]
=> [7,3,5,6,1,2,4] => [4,2,1,6,5,3,7] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(4,3),(5,3),(6,3)],7)
=> ? = 5 - 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [7,3,1,5,2,6,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ? = 4 - 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [6,4,1,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,5),(3,4),(3,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [6,3,1,5,2,7,4] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ? = 4 - 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [6,3,1,4,2,7,5] => ([(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [5,3,1,6,2,7,4] => ([(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(6,5)],7)
=> ? = 3 - 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [5,3,1,4,2,7,6] => ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [7,4,1,6,2,3,5] => ([(1,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7)
=> ? = 4 - 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [7,3,1,6,2,4,5] => ([(1,5),(1,6),(2,3),(2,5),(3,6),(6,4)],7)
=> ? = 4 - 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [7,3,1,5,2,4,6] => ([(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ? = 4 - 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [6,4,1,7,2,3,5] => ([(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(4,5)],7)
=> ? = 3 - 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [6,3,1,7,2,4,5] => ([(0,6),(1,5),(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ? = 3 - 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [5,3,1,7,2,4,6] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(5,4)],7)
=> ? = 4 - 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [6,4,1,5,2,3,7] => ([(0,6),(1,5),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 3 - 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [6,3,1,5,2,4,7] => ([(0,6),(1,4),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 4 - 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [6,3,1,4,2,5,7] => ([(0,4),(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ? = 3 - 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [5,3,1,6,2,4,7] => ([(0,6),(1,5),(1,6),(2,3),(2,6),(3,5),(5,4),(6,4)],7)
=> ? = 3 - 1
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [5,3,1,4,2,6,7] => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ? = 3 - 1
[[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 3 - 1
[[1,4,7],[2],[3],[5],[6]]
=> [6,5,3,2,1,4,7] => [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 3 - 1
[[1,3,7],[2],[4],[5],[6]]
=> [6,5,4,2,1,3,7] => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 3 - 1
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ? = 2 - 1
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [4,1,6,2,7,3,5] => ([(0,3),(0,6),(1,5),(1,6),(2,5),(3,2),(3,4),(6,4)],7)
=> ? = 3 - 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [3,1,6,2,7,4,5] => ([(0,3),(0,6),(1,5),(1,6),(3,4),(3,5),(5,2),(6,4)],7)
=> ? = 3 - 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [4,1,5,2,7,3,6] => ([(0,6),(1,3),(1,6),(2,5),(3,2),(3,4),(6,4),(6,5)],7)
=> ? = 3 - 1
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [3,1,5,2,7,4,6] => ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(5,3),(6,3),(6,4)],7)
=> ? = 4 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000167
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => [.,.]
=> [[]]
=> 1
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [[],[]]
=> 2
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [[[]]]
=> 1
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 3
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 2
[[1,2],[3]]
=> [3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[],[]],[]]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [[[],[],[],[]]]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[]],[[]],[]]
=> 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [[[]],[[],[]]]
=> 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [[[],[]],[[]]]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[],[]],[]]
=> 4
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[[]]],[],[]]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[]],[]],[]]
=> 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> [[[[],[]]],[]]
=> 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [[[[]],[],[]]]
=> 3
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [[[[],[]],[]]]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [[[[],[],[]]]]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> [[[[]]],[[]]]
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[[]],[]],[]]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[[],[]]],[]]
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [[[[]],[[]]]]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [[[[],[]],[]]]
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[[[[]]]],[]]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [[[[[]]],[]]]
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [[[[[]],[]]]]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [[[[[],[]]]]]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [[[[[[]]]]]]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [[],[],[],[],[],[]]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[[]],[],[],[],[]]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [[[],[]],[],[],[]]
=> 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [[[],[],[]],[],[]]
=> 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[],[],[],[]],[]]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> [[[],[],[],[],[]]]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [[[]],[[]],[],[]]
=> 4
[[1,2,4,5,6,7],[3]]
=> [3,1,2,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[],[]],[],[],[],[]]
=> ? = 6
[[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [[[],[],[]],[],[],[]]
=> ? = 6
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [[[],[],[],[]],[],[]]
=> ? = 6
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [[[],[],[],[],[]],[]]
=> ? = 6
[[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[],[]],[],[],[],[]]
=> ? = 6
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [[[]],[[],[]],[],[]]
=> ? = 5
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [[[],[]],[[]],[],[]]
=> ? = 5
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [[[],[],[]],[],[],[]]
=> ? = 6
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [[[]],[[],[],[]],[]]
=> ? = 5
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [[[],[]],[[],[]],[]]
=> ? = 5
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [[[],[],[]],[[]],[]]
=> ? = 5
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [[[],[],[],[]],[],[]]
=> ? = 6
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [[[]],[[],[],[],[]]]
=> ? = 5
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [[[],[]],[[],[],[]]]
=> ? = 5
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [[[],[],[]],[[],[]]]
=> ? = 5
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [[[],[],[],[]],[[]]]
=> ? = 5
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [[[],[],[],[],[]],[]]
=> ? = 6
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [[[[]],[]],[],[],[]]
=> ? = 5
[[1,2,5,6,7],[3],[4]]
=> [4,3,1,2,5,6,7] => [[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [[[[],[]]],[],[],[]]
=> ? = 5
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [[[[]],[],[]],[],[]]
=> ? = 5
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [[[[],[]],[]],[],[]]
=> ? = 5
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
=> [[[[],[],[]]],[],[]]
=> ? = 5
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [[[[]],[],[],[]],[]]
=> ? = 5
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [[[[],[]],[],[]],[]]
=> ? = 5
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [[[.,[.,[.,.]]],[.,.]],[.,.]]
=> [[[[],[],[]],[]],[]]
=> ? = 5
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],.],[.,.]]
=> [[[[],[],[],[]]],[]]
=> ? = 5
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [[[],[]],[],[[]],[]]
=> ? = 5
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [[[]],[[],[]],[],[]]
=> ? = 5
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [[[],[]],[[]],[],[]]
=> ? = 5
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [[[],[],[]],[],[],[]]
=> ? = 6
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [[[]],[[]],[[],[]]]
=> ? = 4
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [[[],[]],[],[[],[]]]
=> ? = 5
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [[[]],[[],[]],[[]]]
=> ? = 4
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [[[],[]],[[]],[[]]]
=> ? = 4
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [[[],[],[]],[],[[]]]
=> ? = 5
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [[[]],[[],[],[]],[]]
=> ? = 5
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [[[],[]],[[],[]],[]]
=> ? = 5
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [[[],[],[]],[[]],[]]
=> ? = 5
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [[[],[],[],[]],[],[]]
=> ? = 6
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [[[[]],[]],[],[],[]]
=> ? = 5
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [[[[],[]]],[],[],[]]
=> ? = 5
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [[[.,.],[[.,.],.]],[.,[.,.]]]
=> [[[[]],[[]]],[],[]]
=> ? = 4
[[1,2,6,7],[3,4],[5]]
=> [5,3,4,1,2,6,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [[[[],[]],[]],[],[]]
=> ? = 5
[[1,4,5,7],[2,6],[3]]
=> [3,2,6,1,4,5,7] => [[[.,.],.],[[.,[.,.]],[.,.]]]
=> [[[[]]],[[],[]],[]]
=> ? = 4
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [[[[]],[]],[[]],[]]
=> ? = 4
[[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [[[.,[.,.]],.],[[.,.],[.,.]]]
=> [[[[],[]]],[[]],[]]
=> ? = 4
[[1,3,4,7],[2,6],[5]]
=> [5,2,6,1,3,4,7] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [[[[]],[],[]],[],[]]
=> ? = 5
[[1,2,4,7],[3,6],[5]]
=> [5,3,6,1,2,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [[[[],[]],[]],[],[]]
=> ? = 5
[[1,2,3,7],[4,6],[5]]
=> [5,4,6,1,2,3,7] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
=> [[[[],[],[]]],[],[]]
=> ? = 5
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [[[[]],[[]],[]],[]]
=> ? = 4
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000245The number of ascents of a permutation. St000542The number of left-to-right-minima of a permutation. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St000155The number of exceedances (also excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St001330The hat guessing number of a graph. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001896The number of right descents of a signed permutations.
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