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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000011
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,0]
=> 1
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [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]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [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]
=> [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]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [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]
=> [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]
=> [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] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 100%●distinct values known / distinct values provided: 83%
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 100%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> => => ? = 1
[[1,2]]
=> 0 => 0 => 2
[[1],[2]]
=> 1 => 1 => 1
[[1,2,3]]
=> 00 => 00 => 3
[[1,3],[2]]
=> 10 => 01 => 2
[[1,2],[3]]
=> 01 => 10 => 1
[[1],[2],[3]]
=> 11 => 11 => 1
[[1,2,3,4]]
=> 000 => 000 => 4
[[1,3,4],[2]]
=> 100 => 001 => 3
[[1,2,4],[3]]
=> 010 => 010 => 2
[[1,2,3],[4]]
=> 001 => 100 => 1
[[1,3],[2,4]]
=> 101 => 101 => 1
[[1,2],[3,4]]
=> 010 => 010 => 2
[[1,4],[2],[3]]
=> 110 => 011 => 2
[[1,3],[2],[4]]
=> 101 => 101 => 1
[[1,2],[3],[4]]
=> 011 => 110 => 1
[[1],[2],[3],[4]]
=> 111 => 111 => 1
[[1,2,3,4,5]]
=> 0000 => 0000 => 5
[[1,3,4,5],[2]]
=> 1000 => 0001 => 4
[[1,2,4,5],[3]]
=> 0100 => 0010 => 3
[[1,2,3,5],[4]]
=> 0010 => 0100 => 2
[[1,2,3,4],[5]]
=> 0001 => 1000 => 1
[[1,3,5],[2,4]]
=> 1010 => 0101 => 2
[[1,2,5],[3,4]]
=> 0100 => 0010 => 3
[[1,3,4],[2,5]]
=> 1001 => 1001 => 1
[[1,2,4],[3,5]]
=> 0101 => 1010 => 1
[[1,2,3],[4,5]]
=> 0010 => 0100 => 2
[[1,4,5],[2],[3]]
=> 1100 => 0011 => 3
[[1,3,5],[2],[4]]
=> 1010 => 0101 => 2
[[1,2,5],[3],[4]]
=> 0110 => 0110 => 2
[[1,3,4],[2],[5]]
=> 1001 => 1001 => 1
[[1,2,4],[3],[5]]
=> 0101 => 1010 => 1
[[1,2,3],[4],[5]]
=> 0011 => 1100 => 1
[[1,4],[2,5],[3]]
=> 1101 => 1011 => 1
[[1,3],[2,5],[4]]
=> 1010 => 0101 => 2
[[1,2],[3,5],[4]]
=> 0110 => 0110 => 2
[[1,3],[2,4],[5]]
=> 1011 => 1101 => 1
[[1,2],[3,4],[5]]
=> 0101 => 1010 => 1
[[1,5],[2],[3],[4]]
=> 1110 => 0111 => 2
[[1,4],[2],[3],[5]]
=> 1101 => 1011 => 1
[[1,3],[2],[4],[5]]
=> 1011 => 1101 => 1
[[1,2],[3],[4],[5]]
=> 0111 => 1110 => 1
[[1],[2],[3],[4],[5]]
=> 1111 => 1111 => 1
[[1,2,3,4,5,6]]
=> 00000 => 00000 => 6
[[1,3,4,5,6],[2]]
=> 10000 => 00001 => 5
[[1,2,4,5,6],[3]]
=> 01000 => 00010 => 4
[[1,2,3,5,6],[4]]
=> 00100 => 00100 => 3
[[1,2,3,4,6],[5]]
=> 00010 => 01000 => 2
[[1,2,3,4,5],[6]]
=> 00001 => 10000 => 1
[[1,3,5,6],[2,4]]
=> 10100 => 00101 => 3
[[1,2,5,6],[3,4]]
=> 01000 => 00010 => 4
[[1,2,3,4,5,6,7,8,9,10,11,12]]
=> 00000000000 => 00000000000 => ? = 12
[]
=> => => ? = 0
[[1,3,4,5,6,7,8,10],[2],[9]]
=> ? => ? => ? = 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [[1]]
=> [1] => 1
[[1,2]]
=> [[1,2]]
=> [2] => 2
[[1],[2]]
=> [[1],[2]]
=> [1,1] => 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => 3
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => 2
[[1,2],[3]]
=> [[1,3],[2]]
=> [1,2] => 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => 4
[[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [3,1] => 3
[[1,2,4],[3]]
=> [[1,2,4],[3]]
=> [2,2] => 2
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [1,3] => 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [1,2,1] => 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [2,2] => 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> [1,2,1] => 1
[[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => 5
[[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> [4,1] => 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3
[[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => 2
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => 2
[[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> [3,2] => 3
[[1,3,4],[2,5]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => 1
[[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> [1,2,2] => 1
[[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => 2
[[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => 2
[[1,3,4],[2],[5]]
=> [[1,3,4],[2],[5]]
=> [1,3,1] => 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [1,2,2] => 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [2,1,2] => 2
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [1,1,2,1] => 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 2
[[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => 1
[[1,3],[2],[4],[5]]
=> [[1,4],[2],[3],[5]]
=> [1,1,2,1] => 1
[[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => 6
[[1,3,4,5,6],[2]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 5
[[1,2,4,5,6],[3]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => 4
[[1,2,3,5,6],[4]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => 3
[[1,2,3,4,6],[5]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => 2
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
[[1,3,5,6],[2,4]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => 3
[[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[[1,5],[2,6],[3,7],[4,8],[9],[10]]
=> [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> [1,1,1,1,1,2,2,1] => ? = 1
[[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> [[1,7,9],[2,8,10],[3],[4],[5],[6]]
=> [1,1,1,1,1,2,2,1] => ? = 1
[[1,3,6],[2,4,7],[5,8],[9],[10]]
=> [[1,6,9],[2,7,10],[3,8],[4],[5]]
=> [1,1,1,1,2,3,1] => ? = 1
[[1,3,7],[2,4,8],[5,9],[6,10]]
=> [[1,5,9],[2,6,10],[3,7],[4,8]]
=> [1,1,1,2,2,2,1] => ? = 1
[[1,4,7],[2,5,8],[3,6,9],[10]]
=> [[1,5,8],[2,6,9],[3,7,10],[4]]
=> [1,1,1,2,3,2] => ? = 1
[[1,2,4,7],[3,5,8],[6,9],[10]]
=> [[1,5,8,10],[2,6,9],[3,7],[4]]
=> [1,1,1,2,3,2] => ? = 1
[[1,3,5,7],[2,4,6,8],[9],[10]]
=> [[1,5,7,9],[2,6,8,10],[3],[4]]
=> [1,1,1,2,2,2,1] => ? = 1
[[1,3,5,8],[2,4,6,9],[7,10]]
=> [[1,4,7,9],[2,5,8,10],[3,6]]
=> [1,1,2,3,2,1] => ? = 1
[[1,2],[3,4],[5,6],[7,9],[8,10]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10]]
=> [1,2,1,2,2,2] => ? = 1
[[1,2],[3,4],[5,7],[6,8],[9,10]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10]]
=> [2,1,2,1,2,2] => ? = 2
[[1,2],[3,4],[5,7],[6,9],[8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> [1,2,2,1,2,2] => ? = 1
[[1,2],[3,5],[4,6],[7,8],[9,10]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10]]
=> [2,2,1,2,1,2] => ? = 2
[[1,2],[3,5],[4,7],[6,9],[8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> [1,2,2,2,1,2] => ? = 1
[[1,2],[3,5],[4,8],[6,9],[7,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> [1,1,2,1,2,1,2] => ? = 1
[[1,2],[3,6],[4,7],[5,9],[8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> [1,2,1,2,2,2] => ? = 1
[[1,3],[2,4],[5,6],[7,9],[8,10]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10]]
=> [1,2,1,2,1,2,1] => ? = 1
[[1,3],[2,4],[5,7],[6,9],[8,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> [1,2,2,1,1,2,1] => ? = 1
[[1,3],[2,4],[5,8],[6,9],[7,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> [1,1,2,2,1,2,1] => ? = 1
[[1,3],[2,5],[4,6],[7,8],[9,10]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10]]
=> [2,2,1,2,2,1] => ? = 2
[[1,3],[2,5],[4,6],[7,9],[8,10]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10]]
=> [1,2,1,1,2,2,1] => ? = 1
[[1,3],[2,5],[4,7],[6,8],[9,10]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10]]
=> [2,1,2,2,2,1] => ? = 2
[[1,3],[2,5],[4,8],[6,9],[7,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> [1,1,2,1,2,2,1] => ? = 1
[[1,3],[2,6],[4,7],[5,9],[8,10]]
=> [[1,3],[2,6],[4,7],[5,9],[8,10]]
=> [1,2,1,2,1,2,1] => ? = 1
[[1,3],[2,6],[4,8],[5,9],[7,10]]
=> [[1,4],[2,6],[3,7],[5,9],[8,10]]
=> [1,1,2,2,1,2,1] => ? = 1
[[1,3],[2,7],[4,8],[5,9],[6,10]]
=> [[1,5],[2,6],[3,7],[4,9],[8,10]]
=> [1,1,1,2,2,2,1] => ? = 1
[[1,4],[2,5],[3,6],[7,9],[8,10]]
=> [[1,3],[2,4],[5,8],[6,9],[7,10]]
=> [1,2,1,1,1,2,2] => ? = 1
[[1,4],[2,5],[3,7],[6,8],[9,10]]
=> [[1,2],[3,5],[4,8],[6,9],[7,10]]
=> [2,1,2,1,2,2] => ? = 2
[[1,4],[2,5],[3,7],[6,9],[8,10]]
=> [[1,3],[2,5],[4,8],[6,9],[7,10]]
=> [1,2,2,1,2,2] => ? = 1
[[1,4],[2,6],[3,7],[5,9],[8,10]]
=> [[1,3],[2,6],[4,8],[5,9],[7,10]]
=> [1,2,1,2,2,2] => ? = 1
[[1,5],[2,6],[3,7],[4,8],[9,10]]
=> [[1,2],[3,7],[4,8],[5,9],[6,10]]
=> [2,1,1,1,2,2,1] => ? = 2
[[1,5],[2,6],[3,7],[4,9],[8,10]]
=> [[1,3],[2,7],[4,8],[5,9],[6,10]]
=> [1,2,1,1,2,2,1] => ? = 1
[[1,5],[2,6],[3,8],[4,9],[7,10]]
=> [[1,4],[2,7],[3,8],[5,9],[6,10]]
=> [1,1,2,1,2,2,1] => ? = 1
[[1,5],[2,7],[3,8],[4,9],[6,10]]
=> [[1,5],[2,7],[3,8],[4,9],[6,10]]
=> [1,1,1,2,2,2,1] => ? = 1
[]
=> []
=> [0] => ? = 0
[[1,2,4,7,8,9,10],[3,5],[6]]
=> [[1,2,3,4,5,8,10],[6,9],[7]]
=> [5,1,2,2] => ? = 5
[[1,3,4,5,6,7],[2,8,9]]
=> ?
=> ? => ? = 2
[[1,2,5,6,7,8],[3,4],[9]]
=> ?
=> ? => ? = 1
[[1,3,8],[2,5],[4,7],[6,9]]
=> ?
=> ? => ? = 1
[[1,2,5,6,7,8,9],[3,4],[10]]
=> ?
=> ? => ? = 1
[[1,2,9],[3,4],[5,6],[7,8],[10]]
=> ?
=> ? => ? = 1
[[1,8,9],[2],[3],[4],[5],[6],[7],[10]]
=> [[1,3,4],[2],[5],[6],[7],[8],[9],[10]]
=> [1,3,1,1,1,1,1,1] => ? = 1
[[1,2,5,6,7,8,9,10],[3],[4]]
=> [[1,2,3,4,5,6,7,10],[8],[9]]
=> [7,1,2] => ? = 7
[[1,2,7,8,9,10],[3],[4],[5],[6]]
=> [[1,2,3,4,5,10],[6],[7],[8],[9]]
=> [5,1,1,1,2] => ? = 5
[[1,2,8,9,10],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,10],[5],[6],[7],[8],[9]]
=> [4,1,1,1,1,2] => ? = 4
[[1,2,7,8],[3,10],[4],[5],[6],[9]]
=> [[1,2,5,10],[3,4],[6],[7],[8],[9]]
=> [2,3,1,1,1,2] => ? = 2
[[1,2],[3,5],[4],[6],[7],[8],[9]]
=> [[1,6],[2,9],[3],[4],[5],[7],[8]]
=> ? => ? = 1
[[1,2],[3,4],[5,6],[7,9],[8]]
=> [[1,2],[3,5],[4,7],[6,9],[8]]
=> ? => ? = 2
[[1,3,10],[2],[4],[5],[6],[7],[8],[9]]
=> [[1,2,9],[3],[4],[5],[6],[7],[8],[10]]
=> ? => ? = 2
[[1,7],[2,8],[3],[4],[5],[6],[9]]
=> ?
=> ? => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [[1]]
=> => => ? = 1 - 1
[[1,2]]
=> [[1],[2]]
=> 1 => 1 => 1 = 2 - 1
[[1],[2]]
=> [[1,2]]
=> 0 => 0 => 0 = 1 - 1
[[1,2,3]]
=> [[1],[2],[3]]
=> 11 => 11 => 2 = 3 - 1
[[1,3],[2]]
=> [[1,2],[3]]
=> 01 => 10 => 1 = 2 - 1
[[1,2],[3]]
=> [[1,3],[2]]
=> 10 => 01 => 0 = 1 - 1
[[1],[2],[3]]
=> [[1,2,3]]
=> 00 => 00 => 0 = 1 - 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 111 => 111 => 3 = 4 - 1
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 011 => 110 => 2 = 3 - 1
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 101 => 101 => 1 = 2 - 1
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 110 => 011 => 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 010 => 010 => 0 = 1 - 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 101 => 101 => 1 = 2 - 1
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 100 => 1 = 2 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 010 => 010 => 0 = 1 - 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 100 => 001 => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 000 => 000 => 0 = 1 - 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 1111 => 4 = 5 - 1
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0111 => 1110 => 3 = 4 - 1
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 1011 => 1101 => 2 = 3 - 1
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 1101 => 1011 => 1 = 2 - 1
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0111 => 0 = 1 - 1
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 0101 => 1010 => 1 = 2 - 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1011 => 1101 => 2 = 3 - 1
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 0110 => 0110 => 0 = 1 - 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 1010 => 0101 => 0 = 1 - 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 1101 => 1011 => 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0011 => 1100 => 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 0101 => 1010 => 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 1001 => 1001 => 1 = 2 - 1
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 0110 => 0110 => 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1010 => 0101 => 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1100 => 0011 => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 0010 => 0100 => 0 = 1 - 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 1010 => 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1001 => 1001 => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 0100 => 0010 => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1010 => 0101 => 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0001 => 1000 => 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 0010 => 0100 => 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 0100 => 0010 => 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1000 => 0001 => 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0000 => 0000 => 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 11111 => 11111 => 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 01111 => 11110 => 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 10111 => 11101 => 3 = 4 - 1
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 11011 => 11011 => 2 = 3 - 1
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 11101 => 10111 => 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 01111 => 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 01011 => 11010 => 2 = 3 - 1
[[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 10111 => 11101 => 3 = 4 - 1
[[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> 11111111111 => 11111111111 => ? = 12 - 1
[[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4],[2,5],[3,6],[7],[8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,5],[2,6],[3,7],[4,8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3,5],[2,4,6],[7],[8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3,6],[2,4,7],[5,8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3,5,7],[2,4,6,8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4],[2,5],[3,6],[7],[8],[9],[10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3,6],[2,4,7],[5,8],[9],[10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4,7],[2,5,8],[3,6,9],[10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3,5,8],[2,4,6,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,4],[5,7],[6,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,2],[3,4],[5,7],[6,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,4],[5,8],[6,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,5],[4,6],[7,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,2],[3,5],[4,7],[6,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,5],[4,8],[6,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,2],[3,6],[4,7],[5,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,6],[4,8],[5,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,2],[3,7],[4,8],[5,9],[6,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,4],[5,6],[7,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,3],[2,4],[5,6],[7,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,4],[5,7],[6,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,3],[2,4],[5,7],[6,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,4],[5,8],[6,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,5],[4,8],[6,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,3],[2,6],[4,7],[5,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,6],[4,8],[5,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,3],[2,7],[4,8],[5,9],[6,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4],[2,5],[3,8],[6,9],[7,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,4],[2,6],[3,7],[5,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[[1,5],[2,6],[3,7],[4,8],[9,10]]
=> ?
=> ? => ? => ? = 2 - 1
[[1,5],[2,6],[3,7],[4,9],[8,10]]
=> ?
=> ? => ? => ? = 1 - 1
[]
=> []
=> => => ? = 0 - 1
[[1,3,4],[2],[5],[6],[7],[8],[9]]
=> ?
=> ? => ? => ? = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000745
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 92%
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 92%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 1
[[1,2]]
=> [[1],[2]]
=> [[1],[2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> [[1,2]]
=> 1
[[1,2,3]]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[[1,3],[2]]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[[1,2],[3]]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[[1],[2],[3]]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 4
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 2
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 1
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [[1,2],[3,4]]
=> 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> 4
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> [[1,4],[2],[3],[5]]
=> 3
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> 2
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> 1
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> 3
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> 2
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> [[1,3,4],[2],[5]]
=> 2
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 2
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> [[1,3,4],[2,5]]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> 1
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 5
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> [[1,5],[2],[3],[4],[6]]
=> 4
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> [[1,4],[2],[3],[5],[6]]
=> 3
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> [[1,3],[2],[4],[5],[6]]
=> 2
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> [[1,4],[2,6],[3],[5]]
=> 3
[[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ?
=> ?
=> ? = 1
[[1,4],[2,5],[3,6],[7],[8],[9]]
=> ?
=> ?
=> ? = 1
[[1,5],[2,6],[3,7],[4,8],[9]]
=> ?
=> ?
=> ? = 1
[[1,3,5],[2,4,6],[7],[8],[9]]
=> ?
=> ?
=> ? = 1
[[1,3,6],[2,4,7],[5,8],[9]]
=> ?
=> ?
=> ? = 1
[[1,3,5,7],[2,4,6,8],[9]]
=> ?
=> ?
=> ? = 1
[[1,4],[2,5],[3,6],[7],[8],[9],[10]]
=> ?
=> ?
=> ? = 1
[[1,3,6],[2,4,7],[5,8],[9],[10]]
=> ?
=> ?
=> ? = 1
[[1,4,7],[2,5,8],[3,6,9],[10]]
=> ?
=> ?
=> ? = 1
[[1,3,5,8],[2,4,6,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,4],[5,7],[6,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,2],[3,4],[5,7],[6,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,4],[5,8],[6,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,5],[4,6],[7,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,2],[3,5],[4,7],[6,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,5],[4,8],[6,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,2],[3,6],[4,7],[5,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,6],[4,8],[5,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,2],[3,7],[4,8],[5,9],[6,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,4],[5,6],[7,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,3],[2,4],[5,6],[7,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,4],[5,7],[6,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,3],[2,4],[5,7],[6,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,4],[5,8],[6,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,5],[4,8],[6,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,3],[2,6],[4,7],[5,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,6],[4,8],[5,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,3],[2,7],[4,8],[5,9],[6,10]]
=> ?
=> ?
=> ? = 1
[[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,4],[2,5],[3,8],[6,9],[7,10]]
=> ?
=> ?
=> ? = 1
[[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,4],[2,6],[3,7],[5,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,5],[2,6],[3,7],[4,8],[9,10]]
=> ?
=> ?
=> ? = 2
[[1,5],[2,6],[3,7],[4,9],[8,10]]
=> ?
=> ?
=> ? = 1
[[1,2,3,4,5,6,7,9],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7],[9]]
=> [[1,3],[2],[4],[5],[6],[7],[8],[9]]
=> ? = 2
[[1,2,3,9],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3],[9]]
=> [[1,3,4,5,6,7],[2],[8],[9]]
=> ? = 2
[[1,2,3,4,5,6,7,8,10],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8],[10]]
=> [[1,3],[2],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000439
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 83%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> [[1]]
=> [1] => [1,0]
=> 2 = 1 + 1
[[1,2]]
=> [[1,2]]
=> [2] => [1,1,0,0]
=> 3 = 2 + 1
[[1],[2]]
=> [[1],[2]]
=> [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3]]
=> [[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,2,4],[3]]
=> [[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3,4],[2,5]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> [[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[1,3],[2],[4],[5]]
=> [[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,3,4,5,6],[2]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[[1,2,4,5,6],[3]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2,3,4,6],[5]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,2,3,5,6,7,8],[4]]
=> [[1,2,3,4,5,7,8],[6]]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[[1,2,3,4,5,7,8],[6]]
=> [[1,2,3,5,6,7,8],[4]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,3,5,6,7,8],[2,4]]
=> [[1,2,3,4,5,7],[6,8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[1,3,4,6,7,8],[2,5]]
=> [[1,2,3,4,6,7],[5,8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,4,5,8],[6,7]]
=> [[1,2,3,6,7,8],[4,5]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,3,5,6,7,8],[2],[4]]
=> [[1,2,3,4,5,7],[6],[8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[1,2,5,6,7,8],[3],[4]]
=> [[1,2,3,4,5,8],[6],[7]]
=> [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[1,3,4,6,7,8],[2],[5]]
=> [[1,2,3,4,6,7],[5],[8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[[1,2,3,6,7,8],[4],[5]]
=> [[1,2,3,4,7,8],[5],[6]]
=> [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 + 1
[[1,2,4,5,7,8],[3],[6]]
=> [[1,2,3,5,6,8],[4],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5,7,8],[4],[6]]
=> [[1,2,3,5,7,8],[4],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,4,7,8],[5],[6]]
=> [[1,2,3,6,7,8],[4],[5]]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[1,2,3,4,5],[6,7,8]]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,3,6,7,8],[2,5],[4]]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[[1,2,6,7,8],[3,5],[4]]
=> [[1,2,3,4,5],[6,8],[7]]
=> [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[1,3,6,7,8],[2,4],[5]]
=> [[1,2,3,4,7],[5,8],[6]]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,3,5,7,8],[2,4],[6]]
=> [[1,2,3,5,7],[4,8],[6]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,4,7,8],[3,5],[6]]
=> [[1,2,3,6,8],[4,7],[5]]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,4,5,8],[3,7],[6]]
=> [[1,2,3,6,8],[4,5],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5,8],[4,7],[6]]
=> [[1,2,3,7,8],[4,5],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,4,8],[5,7],[6]]
=> [[1,2,3,7,8],[4,6],[5]]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[1,5,6,7,8],[2],[3],[4]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[[1,4,6,7,8],[2],[3],[5]]
=> [[1,2,3,4,6],[5],[7],[8]]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[[1,3,6,7,8],[2],[4],[5]]
=> [[1,2,3,4,7],[5],[6],[8]]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,2,6,7,8],[3],[4],[5]]
=> [[1,2,3,4,8],[5],[6],[7]]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 + 1
[[1,4,5,7,8],[2],[3],[6]]
=> [[1,2,3,5,6],[4],[7],[8]]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,3,5,7,8],[2],[4],[6]]
=> [[1,2,3,5,7],[4],[6],[8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,5,7,8],[3],[4],[6]]
=> [[1,2,3,5,8],[4],[6],[7]]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,7,8],[4],[5],[6]]
=> [[1,2,3,7,8],[4],[5],[6]]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,3,4,8],[2,5,7],[6]]
=> [[1,2,3,7],[4,6,8],[5]]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,2,4,8],[3,5,7],[6]]
=> [[1,2,3,6],[4,7,8],[5]]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,4,5],[3,7,8],[6]]
=> [[1,2,3,8],[4,5,6],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,2,3,5],[4,7,8],[6]]
=> [[1,2,3,8],[4,5,7],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[[1,2,3,4],[5,7,8],[6]]
=> [[1,2,3,8],[4,6,7],[5]]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[[1,3,7,8],[2,4],[5,6]]
=> [[1,2,3,4],[5,7],[6,8]]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[[1,2,5,8],[3,4],[6,7]]
=> [[1,2,3,6],[4,5],[7,8]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,3,4,8],[2,5],[6,7]]
=> [[1,2,3,7],[4,6],[5,8]]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,4,7,8],[2,5],[3],[6]]
=> [[1,2,3,6],[4,7],[5],[8]]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,3,7,8],[2,5],[4],[6]]
=> [[1,2,3,5],[4,7],[6],[8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,7,8],[3,5],[4],[6]]
=> [[1,2,3,5],[4,8],[6],[7]]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,3,7,8],[2,4],[5],[6]]
=> [[1,2,3,7],[4,8],[5],[6]]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,4,5,8],[2,7],[3],[6]]
=> [[1,2,3,6],[4,5],[7],[8]]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[[1,3,5,8],[2,7],[4],[6]]
=> [[1,2,3,7],[4,5],[6],[8]]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[[1,2,5,8],[3,7],[4],[6]]
=> [[1,2,3,8],[4,5],[6],[7]]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[[1,3,4,8],[2,7],[5],[6]]
=> [[1,2,3,7],[4,6],[5],[8]]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,2,4,8],[3,7],[5],[6]]
=> [[1,2,3,8],[4,6],[5],[7]]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[[1,6,7,8],[2],[3],[4],[5]]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[[1,5,7,8],[2],[3],[4],[6]]
=> [[1,2,3,5],[4],[6],[7],[8]]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[[1,4,7,8],[2],[3],[5],[6]]
=> [[1,2,3,6],[4],[5],[7],[8]]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000383
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Values
[[1]]
=> [[1]]
=> [1] => [1] => 1
[[1,2]]
=> [[1,2]]
=> [2] => [2] => 2
[[1],[2]]
=> [[1],[2]]
=> [1,1] => [1,1] => 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => [3] => 3
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => [1,2] => 2
[[1,2],[3]]
=> [[1,3],[2]]
=> [1,2] => [2,1] => 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => [4] => 4
[[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [3,1] => [1,3] => 3
[[1,2,4],[3]]
=> [[1,2,4],[3]]
=> [2,2] => [2,2] => 2
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [1,3] => [3,1] => 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [1,2,1] => [1,2,1] => 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [2,2] => [2,2] => 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,1,2] => 2
[[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> [1,2,1] => [1,2,1] => 1
[[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => [5] => 5
[[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,4] => 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => [2,3] => 3
[[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => [3,2] => 2
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [1,4] => [4,1] => 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => [1,2,2] => 2
[[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> [3,2] => [2,3] => 3
[[1,3,4],[2,5]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => [1,3,1] => 1
[[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> [1,2,2] => [2,2,1] => 1
[[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> [2,3] => [3,2] => 2
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,3] => 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => [1,2,2] => 2
[[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => [2,1,2] => 2
[[1,3,4],[2],[5]]
=> [[1,3,4],[2],[5]]
=> [1,3,1] => [1,3,1] => 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [1,2,2] => [2,2,1] => 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [1,2,2] => 2
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [2,1,2] => 2
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,2,1,1] => 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [2,2,1] => 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,1,2] => 2
[[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,1,2,1] => 1
[[1,3],[2],[4],[5]]
=> [[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,2,1,1] => 1
[[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [2,1,1,1] => 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [6] => 6
[[1,3,4,5,6],[2]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,5] => 5
[[1,2,4,5,6],[3]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => [2,4] => 4
[[1,2,3,5,6],[4]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => 3
[[1,2,3,4,6],[5]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => [4,2] => 2
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => 1
[[1,3,5,6],[2,4]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => [1,2,3] => 3
[[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [8] => ? = 8
[[1,2,4,5,6,7,8],[3]]
=> [[1,2,3,4,5,6,8],[7]]
=> [6,2] => [2,6] => ? = 6
[[1,2,3,5,6,7,8],[4]]
=> [[1,2,3,4,5,7,8],[6]]
=> [5,3] => [3,5] => ? = 5
[[1,2,3,4,5,7,8],[6]]
=> [[1,2,3,5,6,7,8],[4]]
=> [3,5] => [5,3] => ? = 3
[[1,2,3,4,5,6,8],[7]]
=> [[1,2,4,5,6,7,8],[3]]
=> [2,6] => [6,2] => ? = 2
[[1,3,5,6,7,8],[2,4]]
=> [[1,2,3,4,5,7],[6,8]]
=> [5,2,1] => [1,2,5] => ? = 5
[[1,2,5,6,7,8],[3,4]]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [2,6] => ? = 6
[[1,3,4,6,7,8],[2,5]]
=> [[1,2,3,4,6,7],[5,8]]
=> [4,3,1] => [1,3,4] => ? = 4
[[1,2,3,4,6,8],[5,7]]
=> [[1,2,4,6,7,8],[3,5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,4,5,8],[6,7]]
=> [[1,2,3,6,7,8],[4,5]]
=> [3,5] => [5,3] => ? = 3
[[1,2,4,5,6,7],[3,8]]
=> [[1,3,4,5,6,8],[2,7]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,3,4,5,6],[7,8]]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => [6,2] => ? = 2
[[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,1,6] => ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [[1,2,3,4,5,7],[6],[8]]
=> [5,2,1] => [1,2,5] => ? = 5
[[1,2,5,6,7,8],[3],[4]]
=> [[1,2,3,4,5,8],[6],[7]]
=> [5,1,2] => [2,1,5] => ? = 5
[[1,3,4,6,7,8],[2],[5]]
=> [[1,2,3,4,6,7],[5],[8]]
=> [4,3,1] => [1,3,4] => ? = 4
[[1,2,4,6,7,8],[3],[5]]
=> [[1,2,3,4,6,8],[5],[7]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,3,6,7,8],[4],[5]]
=> [[1,2,3,4,7,8],[5],[6]]
=> [4,1,3] => [3,1,4] => ? = 4
[[1,2,3,4,7,8],[5],[6]]
=> [[1,2,3,6,7,8],[4],[5]]
=> [3,1,4] => [4,1,3] => ? = 3
[[1,2,3,4,6,8],[5],[7]]
=> [[1,2,4,6,7,8],[3],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,4,5,8],[6],[7]]
=> [[1,2,5,6,7,8],[3],[4]]
=> [2,1,5] => [5,1,2] => ? = 2
[[1,2,4,5,6,7],[3],[8]]
=> [[1,3,4,5,6,8],[2],[7]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,4,7,8],[3,5,6]]
=> [[1,2,3,4,6],[5,7,8]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,5,6,7],[3,4,8]]
=> [[1,3,4,5,6],[2,7,8]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,3,4,5],[6,7,8]]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => [5,3] => ? = 3
[[1,3,6,7,8],[2,5],[4]]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,2,5] => ? = 5
[[1,2,6,7,8],[3,5],[4]]
=> [[1,2,3,4,5],[6,8],[7]]
=> [5,1,2] => [2,1,5] => ? = 5
[[1,2,6,7,8],[3,4],[5]]
=> [[1,2,3,4,6],[5,8],[7]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,2,3,4,8],[5,7],[6]]
=> [[1,2,3,7,8],[4,6],[5]]
=> [3,1,4] => [4,1,3] => ? = 3
[[1,2,3,4,8],[5,6],[7]]
=> [[1,2,4,7,8],[3,6],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,4,6],[5,8],[7]]
=> [[1,2,6,7,8],[3,4],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,3,4,5],[6,8],[7]]
=> [[1,2,6,7,8],[3,5],[4]]
=> [2,1,5] => [5,1,2] => ? = 2
[[1,2,5,6,7],[3,4],[8]]
=> [[1,3,4,5,6],[2,8],[7]]
=> [1,5,2] => [2,5,1] => ? = 1
[[1,2,4,5,6],[3,7],[8]]
=> [[1,4,5,6,8],[2,7],[3]]
=> [1,1,4,2] => [2,4,1,1] => ? = 1
[[1,2,3,4,6],[5,7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [1,1,2,4] => [4,2,1,1] => ? = 1
[[1,5,6,7,8],[2],[3],[4]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [1,1,1,5] => ? = 5
[[1,4,6,7,8],[2],[3],[5]]
=> [[1,2,3,4,6],[5],[7],[8]]
=> [4,2,1,1] => [1,1,2,4] => ? = 4
[[1,2,6,7,8],[3],[4],[5]]
=> [[1,2,3,4,8],[5],[6],[7]]
=> [4,1,1,2] => [2,1,1,4] => ? = 4
[[1,4,5,6,8],[2],[3],[7]]
=> [[1,2,4,5,6],[3],[7],[8]]
=> [2,4,1,1] => [1,1,4,2] => ? = 2
[[1,2,3,4,8],[5],[6],[7]]
=> [[1,2,6,7,8],[3],[4],[5]]
=> [2,1,1,4] => [4,1,1,2] => ? = 2
[[1,2,3,4,6],[5],[7],[8]]
=> [[1,4,6,7,8],[2],[3],[5]]
=> [1,1,2,4] => [4,2,1,1] => ? = 1
[[1,4,5,6],[2,7,8],[3]]
=> [[1,2,5,6],[3,4,7],[8]]
=> [2,4,1,1] => [1,1,4,2] => ? = 2
[[1,2,3,4],[5,7,8],[6]]
=> [[1,2,3,8],[4,6,7],[5]]
=> [3,1,4] => [4,1,3] => ? = 3
[[1,2,3,4],[5,6,8],[7]]
=> [[1,2,4,8],[3,6,7],[5]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,2,7,8],[3,4],[5,6]]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => [2,2,4] => ? = 4
[[1,4,5,6],[2,7],[3,8]]
=> [[1,4,5,6],[2,7],[3,8]]
=> [1,1,4,2] => [2,4,1,1] => ? = 1
[[1,2,3,6],[4,7],[5,8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [1,1,2,4] => [4,2,1,1] => ? = 1
[[1,2,3,4],[5,6],[7,8]]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => [4,2,2] => ? = 2
[[1,4,7,8],[2,5],[3],[6]]
=> [[1,2,3,6],[4,7],[5],[8]]
=> [3,1,2,1,1] => [1,1,2,1,3] => ? = 3
[[1,3,7,8],[2,4],[5],[6]]
=> [[1,2,3,7],[4,8],[5],[6]]
=> [3,1,1,2,1] => [1,2,1,1,3] => ? = 3
Description
The last part of an integer composition.
Matching statistic: St001050
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 64%●distinct values known / distinct values provided: 58%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 64%●distinct values known / distinct values provided: 58%
Values
[[1]]
=> [1] => [1,0]
=> {{1}}
=> 1
[[1,2]]
=> [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 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]
=> {{1},{2},{3},{4},{5},{6}}
=> 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]
=> {{1,2},{3},{4},{5},{6}}
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 4
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 3
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6}}
=> 3
[[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 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]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 6
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6},{7},{8}}
=> ? = 5
[[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]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 6
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,3,4,5},{2},{6},{7},{8}}
=> ? = 4
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 2
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 2
[[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]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,4,7,8],[3,5,6]]
=> [3,5,6,1,2,4,7,8] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> {{1,2,4,6},{3},{5},{7},{8}}
=> ? = 4
[[1,2,5,6,7],[3,4,8]]
=> [3,4,8,1,2,5,6,7] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 1
[[1,3,4,6,7],[2,5,8]]
=> [2,5,8,1,3,4,6,7] => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,3,4,6,7,8},{2},{5}}
=> ? = 1
[[1,2,3,6,7],[4,5,8]]
=> [4,5,8,1,2,3,6,7] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 1
[[1,3,4,5,7],[2,6,8]]
=> [2,6,8,1,3,4,5,7] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,3,4,5,7,8},{2},{6}}
=> ? = 1
[[1,2,3,5,7],[4,6,8]]
=> [4,6,8,1,2,3,5,7] => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 1
[[1,2,3,4,7],[5,6,8]]
=> [5,6,8,1,2,3,4,7] => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[[1,3,4,5,6],[2,7,8]]
=> [2,7,8,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,3,4,5,6,8},{2},{7}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [3,7,8,1,2,4,5,6] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,4,5,6,8},{3},{7}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,5},{3,4},{6},{7},{8}}
=> ? = 5
[[1,2,6,7,8],[3,5],[4]]
=> [4,3,5,1,2,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,5},{3,4},{6},{7},{8}}
=> ? = 5
[[1,3,6,7,8],[2,4],[5]]
=> [5,2,4,1,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,2,6,7,8],[3,4],[5]]
=> [5,3,4,1,2,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,3,5,7,8],[2,4],[6]]
=> [6,2,4,1,3,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,4,7,8],[3,5],[6]]
=> [6,3,5,1,2,4,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,3,4,6,8],[2,7],[5]]
=> [5,2,7,1,3,4,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 2
[[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 2
[[1,2,3,6,8],[4,7],[5]]
=> [5,4,7,1,2,3,6,8] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 2
[[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[[1,4,6,7,8],[2],[3],[5]]
=> [5,3,2,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,3,6,7,8],[2],[4],[5]]
=> [5,4,2,1,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,2,6,7,8],[3],[4],[5]]
=> [5,4,3,1,2,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[[1,4,5,7,8],[2],[3],[6]]
=> [6,3,2,1,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,3,5,7,8],[2],[4],[6]]
=> [6,4,2,1,3,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,5,7,8],[3],[4],[6]]
=> [6,4,3,1,2,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,3,7,8],[4],[5],[6]]
=> [6,5,4,1,2,3,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 3
[[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> {{1,2,5,7,8},{3},{4},{6}}
=> ? = 1
[[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,4,7,8},{3},{5},{6}}
=> ? = 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000678
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => [1,0]
=> ? = 1
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [2,4,1,3,5,6,7,8] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [1,3,2,4,7,5,6,8] => ?
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [4,5,8,1,2,3,6,7] => [1,2,4,5,3,6,8,7] => ?
=> ? = 1
[[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,4,1,5,3,6,7,8] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[[1,3,4,6,8],[2,7],[5]]
=> [5,2,7,1,3,4,6,8] => [2,1,5,3,4,7,6,8] => ?
=> ? = 2
[[1,2,3,6,8],[4,7],[5]]
=> [5,4,7,1,2,3,6,8] => [1,2,5,4,3,7,6,8] => ?
=> ? = 2
[[1,3,4,5,8],[2,7],[6]]
=> [6,2,7,1,3,4,5,8] => [2,1,3,6,4,7,5,8] => ?
=> ? = 3
[[1,2,5,6,8],[3,4],[7]]
=> [7,3,4,1,2,5,6,8] => [1,3,4,2,7,5,6,8] => ?
=> ? = 2
[[1,2,3,5,8],[4,6],[7]]
=> [7,4,6,1,2,3,5,8] => [1,2,4,3,7,6,5,8] => ?
=> ? = 2
[[1,4,5,6,7],[2,8],[3]]
=> [3,2,8,1,4,5,6,7] => [3,2,1,4,5,6,8,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[1,3,5,6,7],[2,8],[4]]
=> [4,2,8,1,3,5,6,7] => [2,4,1,3,5,6,8,7] => ?
=> ? = 1
[[1,2,4,5,7],[3,8],[6]]
=> [6,3,8,1,2,4,5,7] => [1,3,2,6,4,5,8,7] => ?
=> ? = 1
[[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[[1,4,6,7,8],[2],[3],[5]]
=> [5,3,2,1,4,6,7,8] => [3,5,2,1,4,6,7,8] => [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,3,6,7,8],[2],[4],[5]]
=> [5,4,2,1,3,6,7,8] => [2,5,4,1,3,6,7,8] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[[1,4,5,7,8],[2],[3],[6]]
=> [6,3,2,1,4,5,7,8] => [3,2,6,1,4,5,7,8] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[[1,3,5,7,8],[2],[4],[6]]
=> [6,4,2,1,3,5,7,8] => [2,4,6,1,3,5,7,8] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[[1,2,5,7,8],[3],[4],[6]]
=> [6,4,3,1,2,5,7,8] => [1,4,6,3,2,5,7,8] => ?
=> ? = 3
[[1,4,5,6,8],[2],[3],[7]]
=> [7,3,2,1,4,5,6,8] => [3,2,1,7,4,5,6,8] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[[1,4,5,6,7],[2],[3],[8]]
=> [8,3,2,1,4,5,6,7] => [3,2,1,4,8,5,6,7] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,3,5,6,7],[2],[4],[8]]
=> [8,4,2,1,3,5,6,7] => [2,4,1,3,8,5,6,7] => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,3,4,8],[2,5,7],[6]]
=> [6,2,5,7,1,3,4,8] => [2,1,3,6,5,7,4,8] => ?
=> ? = 3
[[1,3,5,8],[2,4,6],[7]]
=> [7,2,4,6,1,3,5,8] => [2,1,4,3,7,6,5,8] => ?
=> ? = 2
[[1,2,5,8],[3,4,6],[7]]
=> [7,3,4,6,1,2,5,8] => [1,3,4,2,7,6,5,8] => ?
=> ? = 2
[[1,2,3,8],[4,5,6],[7]]
=> [7,4,5,6,1,2,3,8] => [1,2,4,5,7,6,3,8] => ?
=> ? = 2
[[1,3,6,7],[2,5,8],[4]]
=> [4,2,5,8,1,3,6,7] => [2,4,1,5,3,6,8,7] => [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 1
[[1,2,6,7],[3,5,8],[4]]
=> [4,3,5,8,1,2,6,7] => [1,4,3,5,2,6,8,7] => ?
=> ? = 1
[[1,3,5,7],[2,6,8],[4]]
=> [4,2,6,8,1,3,5,7] => [2,4,1,3,6,5,8,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[1,2,5,7],[3,6,8],[4]]
=> [4,3,6,8,1,2,5,7] => [1,4,3,2,6,5,8,7] => ?
=> ? = 1
[[1,3,4,7],[2,6,8],[5]]
=> [5,2,6,8,1,3,4,7] => ? => ?
=> ? = 1
[[1,2,3,7],[4,5,8],[6]]
=> [6,4,5,8,1,2,3,7] => [1,2,4,6,5,3,8,7] => ?
=> ? = 1
[[1,4,5,6],[2,7,8],[3]]
=> [3,2,7,8,1,4,5,6] => [3,2,1,4,5,7,8,6] => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[[1,3,4,5],[2,7,8],[6]]
=> [6,2,7,8,1,3,4,5] => [2,1,3,6,4,7,8,5] => ?
=> ? = 3
[[1,2,5,7],[3,4,6],[8]]
=> [8,3,4,6,1,2,5,7] => [1,3,4,2,6,8,5,7] => ?
=> ? = 1
[[1,2,4,7],[3,5,6],[8]]
=> [8,3,5,6,1,2,4,7] => [1,3,2,5,6,8,4,7] => ?
=> ? = 1
[[1,3,4,5],[2,6,7],[8]]
=> [8,2,6,7,1,3,4,5] => [2,1,3,4,6,8,7,5] => ?
=> ? = 1
[[1,4,6,8],[2,5],[3,7]]
=> [3,7,2,5,1,4,6,8] => [3,2,1,5,7,4,6,8] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2
[[1,2,6,8],[3,5],[4,7]]
=> [4,7,3,5,1,2,6,8] => [1,4,3,5,7,2,6,8] => ?
=> ? = 2
[[1,3,6,8],[2,4],[5,7]]
=> [5,7,2,4,1,3,6,8] => [2,1,5,4,7,3,6,8] => ?
=> ? = 2
[[1,4,5,8],[2,6],[3,7]]
=> [3,7,2,6,1,4,5,8] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[1,3,4,8],[2,6],[5,7]]
=> [5,7,2,6,1,3,4,8] => [2,1,5,3,7,6,4,8] => ?
=> ? = 2
[[1,3,6,7],[2,5],[4,8]]
=> [4,8,2,5,1,3,6,7] => [2,4,1,5,3,8,6,7] => [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,6,7],[2,4],[5,8]]
=> [5,8,2,4,1,3,6,7] => [2,1,5,4,3,8,6,7] => ?
=> ? = 1
[[1,2,6,7],[3,4],[5,8]]
=> [5,8,3,4,1,2,6,7] => [1,3,5,4,2,8,6,7] => ?
=> ? = 1
[[1,4,5,7],[2,6],[3,8]]
=> [3,8,2,6,1,4,5,7] => [3,2,1,4,6,8,5,7] => [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1
[[1,3,5,7],[2,6],[4,8]]
=> [4,8,2,6,1,3,5,7] => [2,4,1,3,6,8,5,7] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 1
[[1,2,3,7],[4,6],[5,8]]
=> [5,8,4,6,1,2,3,7] => [1,2,5,4,6,8,3,7] => ?
=> ? = 1
[[1,2,5,7],[3,4],[6,8]]
=> [6,8,3,4,1,2,5,7] => [1,3,4,6,2,8,5,7] => ?
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000025
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 58%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 58%
Values
[[1]]
=> [[1]]
=> [1] => [1,0]
=> 1
[[1,2]]
=> [[1,2]]
=> [2] => [1,1,0,0]
=> 2
[[1],[2]]
=> [[1],[2]]
=> [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 3
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[1,2,4],[3]]
=> [[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,3,4],[2,5]]
=> [[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3,4],[2],[5]]
=> [[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[[1,3],[2],[4],[5]]
=> [[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,3,4,5,6],[2]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,2,4,5,6],[3]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[[1,2,3,5,6],[4]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,3,5,6],[2,4]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [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]]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,4,5,6,7,8],[3]]
=> [[1,2,3,4,5,6,8],[7]]
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,2,3,5,6,7,8],[4]]
=> [[1,2,3,4,5,7,8],[6]]
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[[1,2,3,4,6,7,8],[5]]
=> [[1,2,3,4,6,7,8],[5]]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,2,3,4,5,7,8],[6]]
=> [[1,2,3,5,6,7,8],[4]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[[1,2,3,4,5,6,8],[7]]
=> [[1,2,4,5,6,7,8],[3]]
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,2,3,4,5,6,7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[[1,3,5,6,7,8],[2,4]]
=> [[1,2,3,4,5,7],[6,8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,5,6,7,8],[3,4]]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,3,4,6,7,8],[2,5]]
=> [[1,2,3,4,6,7],[5,8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[1,3,4,5,6,8],[2,7]]
=> [[1,2,4,5,6,7],[3,8]]
=> [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,6,8],[3,7]]
=> [[1,2,4,5,6,8],[3,7]]
=> [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[1,2,3,4,6,8],[5,7]]
=> [[1,2,4,6,7,8],[3,5]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[[1,2,3,4,5,8],[6,7]]
=> [[1,2,3,6,7,8],[4,5]]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[[1,3,4,5,6,7],[2,8]]
=> [[1,3,4,5,6,7],[2,8]]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,5,6,7],[3,8]]
=> [[1,3,4,5,6,8],[2,7]]
=> [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,5,6,7],[4,8]]
=> [[1,3,4,5,7,8],[2,6]]
=> [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,2,3,4,6,7],[5,8]]
=> [[1,3,4,6,7,8],[2,5]]
=> [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,2,3,4,5,7],[6,8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,2,3,4,5,6],[7,8]]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,5,6,7,8],[2],[4]]
=> [[1,2,3,4,5,7],[6],[8]]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[[1,2,5,6,7,8],[3],[4]]
=> [[1,2,3,4,5,8],[6],[7]]
=> [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[[1,3,4,6,7,8],[2],[5]]
=> [[1,2,3,4,6,7],[5],[8]]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[[1,2,4,6,7,8],[3],[5]]
=> [[1,2,3,4,6,8],[5],[7]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,6,7,8],[4],[5]]
=> [[1,2,3,4,7,8],[5],[6]]
=> [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[1,3,4,5,7,8],[2],[6]]
=> [[1,2,3,5,6,7],[4],[8]]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3
[[1,2,4,5,7,8],[3],[6]]
=> [[1,2,3,5,6,8],[4],[7]]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[[1,2,3,5,7,8],[4],[6]]
=> [[1,2,3,5,7,8],[4],[6]]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[1,2,3,4,7,8],[5],[6]]
=> [[1,2,3,6,7,8],[4],[5]]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[1,3,4,5,6,8],[2],[7]]
=> [[1,2,4,5,6,7],[3],[8]]
=> [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[[1,2,4,5,6,8],[3],[7]]
=> [[1,2,4,5,6,8],[3],[7]]
=> [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[1,2,3,5,6,8],[4],[7]]
=> [[1,2,4,5,7,8],[3],[6]]
=> [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[[1,2,3,4,6,8],[5],[7]]
=> [[1,2,4,6,7,8],[3],[5]]
=> [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[[1,2,3,4,5,8],[6],[7]]
=> [[1,2,5,6,7,8],[3],[4]]
=> [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[[1,3,4,5,6,7],[2],[8]]
=> [[1,3,4,5,6,7],[2],[8]]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[[1,2,4,5,6,7],[3],[8]]
=> [[1,3,4,5,6,8],[2],[7]]
=> [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[1,2,3,5,6,7],[4],[8]]
=> [[1,3,4,5,7,8],[2],[6]]
=> [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,2,3,4,6,7],[5],[8]]
=> [[1,3,4,6,7,8],[2],[5]]
=> [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[1,2,3,4,5,7],[6],[8]]
=> [[1,3,5,6,7,8],[2],[4]]
=> [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[1,2,3,4,5,6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[[1,2,4,7,8],[3,5,6]]
=> [[1,2,3,4,6],[5,7,8]]
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[1,2,3,4,8],[5,6,7]]
=> [[1,2,3,4,8],[5,6,7]]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[1,3,5,6,7],[2,4,8]]
=> [[1,3,4,5,7],[2,6,8]]
=> [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[[1,2,5,6,7],[3,4,8]]
=> [[1,3,4,5,6],[2,7,8]]
=> [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[[1,3,4,6,7],[2,5,8]]
=> [[1,3,4,6,7],[2,5,8]]
=> [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[[1,2,3,6,7],[4,5,8]]
=> [[1,3,4,5,8],[2,6,7]]
=> [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[1,3,4,5,7],[2,6,8]]
=> [[1,3,5,6,7],[2,4,8]]
=> [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[[1,2,3,5,7],[4,6,8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000026The position of the first return of a Dyck path. St000989The number of final rises of a permutation. St000990The first ascent of a permutation. St000654The first descent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000260The radius of a connected graph. St000993The multiplicity of the largest part of an integer partition.
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