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Your data matches 117 different statistics following compositions of up to 3 maps.
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Matching statistic: St000201
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [.,.]
=> 1
[1,0,1,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 1
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size n, at least 1, with exactly one leaf node for is 2n−1, see [2].
The number of binary tree of size n, at least 3, with exactly two leaf nodes is n(n+1)2n−2, see [3].
Matching statistic: St000390
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 1
[1,0,1,0]
=> [1,1] => [1,1] => 11 => 1
[1,1,0,0]
=> [2] => [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 111 => 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => 101 => 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => 110 => 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => 110 => 1
[1,1,1,0,0,0]
=> [3] => [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1111 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 1011 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => 1101 => 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => 1101 => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1001 => 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 1110 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1110 => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => 1110 => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 1010 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 1100 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 1100 => 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 11111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 10111 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 11011 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 11011 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 10011 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => 11001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => 11001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => 11001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 10001 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 10010 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => 10010 => 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [1,9] => 1100000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => [9,1] => 1000000001 => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [1,10] => 11000000000 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [1,1,8] => 1110000000 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => [1,8,1] => 1100000001 => ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => [10,1] => 10000000001 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3] => 1111111100 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,2,2,1] => [1,2,2,2,2,1] => 1101010101 => ? = 5
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,1,2,1] => [1,2,1,3,2,1] => 1101100101 => ? = 4
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,1,2,2,1] => [1,2,2,1,3,1] => 1101011001 => ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,3,2,1,2,1] => [1,2,1,2,3,1] => 1101101001 => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,1,1,2,1] => [1,2,1,1,4,1] => 1101110001 => ? = 3
[1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,3,2,2,1,1] => [1,1,2,2,3,1] => 1110101001 => ? = 4
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,3,3,1,1,1] => [1,1,1,3,3,1] => 1111001001 => ? = 3
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,1,2,1,1] => [1,1,2,1,4,1] => 1110110001 => ? = 3
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,4,2,1,1,1] => [1,1,1,2,4,1] => 1111010001 => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,5,1,1,1,1] => [1,1,1,1,5,1] => 1111100001 => ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,2,2,2,1] => [1,2,2,2,1,2] => 1101010110 => ? = 4
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [2,1,3,1,2,1] => [1,2,1,3,1,2] => 1101100110 => ? = 3
[1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,1,3,2,1,1] => [1,1,2,3,1,2] => 1110100110 => ? = 3
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,1,4,1,1,1] => [1,1,1,4,1,2] => 1111000110 => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,2,1,2,2,1] => [1,2,2,1,2,2] => 1101011010 => ? = 4
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,2,1,3,1,1] => [1,1,3,1,2,2] => 1110011010 => ? = 3
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,1,1,2,2,1] => [1,2,2,1,1,3] => 1101011100 => ? = 3
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,1,1,3,1,1] => [1,1,3,1,1,3] => 1110011100 => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [2,2,2,1,2,1] => [1,2,1,2,2,2] => 1101101010 => ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,1,2,1] => [1,2,1,1,3,2] => 1101110010 => ? = 3
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,1,2,1] => [1,2,1,2,1,3] => 1101101100 => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,1,1,2,1] => [1,2,1,1,2,3] => 1101110100 => ? = 3
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,1,1,1,2,1] => [1,2,1,1,1,4] => 1101111000 => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,1,1,1,1] => [1,1,1,1,1,5] => 1111110000 => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,2,2,2,2,2,1] => [1,2,2,2,2,2,1] => 110101010101 => ? = 6
[1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,3,1,2,1] => [1,2,1,3,2,2,1] => ? => ? = 5
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,3,1,2,2,1] => [1,2,2,1,3,2,1] => ? => ? = 5
[1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,2,1,2,1] => [1,2,1,2,3,2,1] => ? => ? = 5
[1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,4,1,1,2,1] => [1,2,1,1,4,2,1] => ? => ? = 4
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,1,2,2,2,1] => [1,2,2,2,1,3,1] => ? => ? = 5
[1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,1,3,1,2,1] => [1,2,1,3,1,3,1] => ? => ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,2,1,2,2,1] => [1,2,2,1,2,3,1] => 110101101001 => ? = 5
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,1,1,2,2,1] => [1,2,2,1,1,4,1] => ? => ? = 4
[1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,3,2,2,1,2,1] => [1,2,1,2,2,3,1] => ? => ? = 5
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,3,3,1,1,2,1] => [1,2,1,1,3,3,1] => ? => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,4,1,2,1,2,1] => [1,2,1,2,1,4,1] => ? => ? = 4
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,1,1,2,1] => [1,2,1,1,2,4,1] => ? => ? = 4
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,1,1,1,2,1] => [1,2,1,1,1,5,1] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,3,2,2,2,1,1] => [1,1,2,2,2,3,1] => 111010101001 => ? = 5
[1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,3,2,3,1,1,1] => [1,1,1,3,2,3,1] => ? => ? = 4
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,3,3,1,2,1,1] => [1,1,2,1,3,3,1] => ? => ? = 4
Description
The number of runs of ones in a binary word.
Matching statistic: St000292
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [1,1] => 11 => 0 = 1 - 1
[1,1,0,0]
=> [2] => [2] => 10 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 111 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => 101 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => 110 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => 110 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [3] => 100 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 1011 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => 1101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => 1101 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1001 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 1110 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1110 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => 1110 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => 1010 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1100 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => 1100 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => 1100 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => 1000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 11111 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 10111 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 11011 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => 11011 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 10011 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => 11101 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 10101 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 10001 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 10010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => 11110 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => 10110 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => 11010 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => 11010 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => 10010 => 1 = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [1,9] => 1100000000 => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => [9,1] => 1000000001 => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [1,10] => 11000000000 => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [1,1,8] => 1110000000 => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => [1,8,1] => 1100000001 => ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => [10,1] => 10000000001 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3] => 1111111100 => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,2,2,1] => [1,2,2,2,2,1] => 1101010101 => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,1,2,1] => [1,2,1,3,2,1] => 1101100101 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,1,2,2,1] => [1,2,2,1,3,1] => 1101011001 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,3,2,1,2,1] => [1,2,1,2,3,1] => 1101101001 => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,1,1,2,1] => [1,2,1,1,4,1] => 1101110001 => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,3,2,2,1,1] => [1,1,2,2,3,1] => 1110101001 => ? = 4 - 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,3,3,1,1,1] => [1,1,1,3,3,1] => 1111001001 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,1,2,1,1] => [1,1,2,1,4,1] => 1110110001 => ? = 3 - 1
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,4,2,1,1,1] => [1,1,1,2,4,1] => 1111010001 => ? = 3 - 1
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,5,1,1,1,1] => [1,1,1,1,5,1] => 1111100001 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,2,2,2,1] => [1,2,2,2,1,2] => 1101010110 => ? = 4 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [2,1,3,1,2,1] => [1,2,1,3,1,2] => 1101100110 => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,1,3,2,1,1] => [1,1,2,3,1,2] => 1110100110 => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,1,4,1,1,1] => [1,1,1,4,1,2] => 1111000110 => ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,2,1,2,2,1] => [1,2,2,1,2,2] => 1101011010 => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,2,1,3,1,1] => [1,1,3,1,2,2] => 1110011010 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,1,1,2,2,1] => [1,2,2,1,1,3] => 1101011100 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,1,1,3,1,1] => [1,1,3,1,1,3] => 1110011100 => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [2,2,2,1,2,1] => [1,2,1,2,2,2] => 1101101010 => ? = 4 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,1,2,1] => [1,2,1,1,3,2] => 1101110010 => ? = 3 - 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,1,2,1] => [1,2,1,2,1,3] => 1101101100 => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,1,1,2,1] => [1,2,1,1,2,3] => 1101110100 => ? = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,1,1,1,2,1] => [1,2,1,1,1,4] => 1101111000 => ? = 2 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,1,1,1,1] => [1,1,1,1,1,5] => 1111110000 => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,2,2,2,2,2,1] => [1,2,2,2,2,2,1] => 110101010101 => ? = 6 - 1
[1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,3,1,2,1] => [1,2,1,3,2,2,1] => ? => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,3,1,2,2,1] => [1,2,2,1,3,2,1] => ? => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,2,1,2,1] => [1,2,1,2,3,2,1] => ? => ? = 5 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,4,1,1,2,1] => [1,2,1,1,4,2,1] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,1,2,2,2,1] => [1,2,2,2,1,3,1] => ? => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,1,3,1,2,1] => [1,2,1,3,1,3,1] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,2,1,2,2,1] => [1,2,2,1,2,3,1] => 110101101001 => ? = 5 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,1,1,2,2,1] => [1,2,2,1,1,4,1] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,3,2,2,1,2,1] => [1,2,1,2,2,3,1] => ? => ? = 5 - 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,3,3,1,1,2,1] => [1,2,1,1,3,3,1] => ? => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,4,1,2,1,2,1] => [1,2,1,2,1,4,1] => ? => ? = 4 - 1
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,2,1,1,2,1] => [1,2,1,1,2,4,1] => ? => ? = 4 - 1
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,1,1,1,2,1] => [1,2,1,1,1,5,1] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,3,2,2,2,1,1] => [1,1,2,2,2,3,1] => 111010101001 => ? = 5 - 1
[1,0,1,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0,0]
=> [1,3,2,3,1,1,1] => [1,1,1,3,2,3,1] => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,3,3,1,2,1,1] => [1,1,2,1,3,3,1] => ? => ? = 4 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000291
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 83%
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => => => ? = 1 - 1
[1,0,1,0]
=> [1,2] => 0 => 0 => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => 1 => 1 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 01 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 10 => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => 11 => 11 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 100 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 010 => 100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 101 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 001 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 110 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 100 => 001 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 100 => 001 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 101 => 110 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 011 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 110 => 011 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 110 => 011 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 0100 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0010 => 0100 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0101 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1000 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1010 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0100 => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0100 => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 1010 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 1001 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 1011 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 0001 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 0110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1100 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1010 => 1100 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1000 => 0001 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1001 => 0110 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1000 => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1000 => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1001 => 0110 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1010 => 1100 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1010 => 1100 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1010 => 1100 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1011 => 1101 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 0011 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,5,3,8,7] => ? => ? => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,2,5,4,6,3,8,7] => ? => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,2,5,4,7,6,3,8] => ? => ? => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,2,4,7,8,6,5] => ? => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,5,8,7,2] => ? => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,5,4,6,8,7,2] => ? => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,5,8,7,4,2] => ? => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,5,8,4,2] => ? => ? => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ? => ? => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,4,3,2,7,8,6,5] => ? => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,4,3,5,6,8,7,2] => ? => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,3,7,8,6,2] => ? => ? => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,4,6,8,7,5] => ? => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,1,5,6,8,7,4] => ? => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,7,8,6,4] => ? => ? => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,4,3,5,7,8,6,1] => ? => ? => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => ? => ? => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,4,3,7,8,6,5,1] => ? => ? => ? = 3 - 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,4,2,6,7,5,1,8] => ? => ? => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> [3,5,4,2,6,8,7,1] => ? => ? => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,5,6,4,2,7,1,8] => ? => ? => ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [4,3,6,5,2,7,1,8] => ? => ? => ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => 0111111111 => 1011111111 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,5,4,7,9,8,6,1] => ? => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,9,10,8,4,2] => ? => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,3,7,9,10,8,6,2] => ? => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,4,6,7,5,3,9,10,8,2] => ? => ? => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,6,4,7,3,9,10,8,2] => ? => ? => ? = 3 - 1
[1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,7,5,3,10,2] => ? => ? => ? = 4 - 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,6,9,5,3,10,2] => ? => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,8,9,7,3,10,2] => ? => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,6,4,9,3,10,2] => ? => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,3,1,5,7,9,10,8,6,4] => ? => ? => ? = 4 - 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,9,10,8,4] => ? => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,6,8,9,7,5,10,4] => ? => ? => ? = 3 - 1
[1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,3,1,7,8,6,9,5,10,4] => ? => ? => ? = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1,7,9,10,8,6] => ? => ? => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,1,8,9,7,10,6] => ? => ? => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,4,2,5,1,7,9,10,8,6] => ? => ? => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1,8,9,7,10,6] => ? => ? => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [2,4,6,7,5,3,1,9,10,8] => ? => ? => ? = 4 - 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [2,5,6,4,7,3,1,9,10,8] => ? => ? => ? = 3 - 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,2,6,7,5,1,9,10,8] => ? => ? => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [3,5,6,4,2,7,1,9,10,8] => ? => ? => ? = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,5,3,6,2,7,1,9,10,8] => ? => ? => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,3,5,7,9,11,12,10,8,6,4,2] => ? => ? => ? = 6 - 1
[1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,8,9,7,11,12,10,6,4,2] => ? => ? => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,6,7,5,9,11,12,10,8,4,2] => ? => ? => ? = 5 - 1
[1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,8,9,7,5,11,12,10,4,2] => ? => ? => ? = 5 - 1
Description
The number of descents of a binary word.
Matching statistic: St000386
(load all 93 compositions to match this statistic)
(load all 93 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
St000386: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001712
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [2,1] => [[1],[2]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => [[1,2],[3]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [[1,4],[2],[3]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => ?
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,6,3,2,1] => [1,2,4,5,3,6,7,8] => ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,4,2,1] => [3,4,2,1,6,5,7,8] => ?
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,6,2,1] => [2,1,4,5,6,3,7,8] => ?
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,6,2,1] => [2,1,5,4,6,3,7,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => [3,4,2,1,5,7,6,8] => ?
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => [2,1,4,5,3,7,6,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => [4,3,2,5,1,7,6,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,2,3,1] => [4,3,5,2,1,7,6,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => [2,1,4,3,6,7,5,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,3,2,8,1] => [3,2,5,4,6,7,1,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => [2,4,5,3,6,7,1,8] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,2,8,1] => [3,2,4,6,5,7,1,8] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,7,2,8,1] => [4,3,6,5,2,7,1,8] => ?
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [7,6,8,5,2,3,4,1] => [2,3,1,4,7,6,5,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,2,3,4,1] => [2,1,4,3,7,6,5,8] => ?
=> ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,2,3,4,1] => [2,4,3,1,7,6,5,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,2,3,4,1] => [3,4,2,1,7,6,5,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,2,3,8,1] => [3,2,4,5,7,6,1,8] => ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [5,6,7,2,3,4,8,1] => [4,3,2,7,6,5,1,8] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,3,1,2] => [3,4,2,1,5,6,8,7] => ?
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,1,2] => [2,3,1,5,4,6,8,7] => ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,1,2] => [3,2,1,5,4,6,8,7] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,1,2] => [3,2,1,4,6,5,8,7] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,3,8,1,2] => [3,2,4,5,6,1,8,7] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => [3,2,4,6,5,1,8,7] => ?
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,1,2] => [2,5,4,6,3,1,8,7] => ?
=> ? = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,1,3] => [3,2,4,5,1,7,8,6] => ?
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,2,3,1,8] => [4,5,3,2,7,6,8,1] => ?
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,1,2,7] => [1,5,4,6,3,8,7,2] => ?
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> [6,7,5,2,3,1,4,8] => [3,2,4,7,6,8,5,1] => ?
=> ? = 3 - 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,1,2,5,7] => [1,3,6,5,8,7,4,2] => ?
=> ? = 2 - 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [7,5,2,3,1,4,6,8] => [2,4,7,6,8,5,3,1] => ?
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [1,8,9,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [7,8,6,5,4,3,2,1,9] => [[1,2,9],[3],[4],[5],[6],[7],[8]]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [9,8,7,6,5,4,3,2,1,10] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [1,9,10,8,7,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,7,9,8,6,5,4,3,2] => [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [1,8,7,9,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9]]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [6,8,7,5,4,3,2,1,9] => [[1,2,9],[3],[4],[5],[6],[7],[8]]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [8,9,7,6,5,4,3,2,1,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,9] => [2,3,4,5,6,7,8,9,1] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,2,3,1,9] => [2,3,4,5,6,8,7,9,1] => [[1,2,3,4,5,6,8],[7],[9]]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,1,2,9] => [2,3,4,5,6,7,9,8,1] => [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 1 - 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
Matching statistic: St001037
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000568
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [.,.]
=> ? = 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,2,4,6,5,3,8,7] => ?
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,2,5,4,6,3,8,7] => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,2,5,4,7,6,3,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,2,4,7,8,6,5] => ?
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,5,8,7,2] => ?
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,5,4,6,8,7,2] => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,5,8,7,4,2] => ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,4,3,2,7,8,6,5] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,4,3,5,6,8,7,2] => ?
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,4,6,8,7,5] => ?
=> ? = 3
[1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,1,5,6,8,7,4] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,4,3,5,7,8,6,1] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,4,3,7,6,8,5,1] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,4,3,7,8,6,5,1] => ?
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,6,8,7] => [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> ? = 2
[1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,5,6,8,7,4] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,1,5,7,8,6,4] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,1,5,8,7,6,4] => [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,6,7,8,5,4] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,2,1,6,8,7,5,4] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 3
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,1,7,8,6,5,4] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5,8,7] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,2,4,1,6,8,7,5] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,2,4,1,7,6,8,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,5,4,1,6,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,5,4,6,1,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,7,6,1,8] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,7,6,8,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,5,7,6,4,1,8] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,2,6,5,7,4,8,1] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,4,2,1,7,8,6,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,2,5,1,7,8,6] => [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,4,2,6,7,5,1,8] => [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
=> ? = 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,2,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> [3,5,4,2,6,8,7,1] => ?
=> ? = 3
[1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [3,5,4,7,6,2,8,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,5,6,4,2,7,1,8] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [4,3,5,2,6,1,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [4,3,5,2,7,6,8,1] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [4,3,6,5,2,7,1,8] => ?
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,3,6,5,7,2,8,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,3,6,2,7,1,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,6,5,3,7,2,8,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St001280
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [4,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => [2,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => [2,2,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => [2,2,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,2,1] => [2,2,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => [3,2,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => [2,2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => [2,2,2]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,2,1,1] => [2,2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,2,1,1] => [2,2,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => [2,2,2]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => [3,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => [3,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => [3,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => [4,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => [3,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => [3,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => [3,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => [3,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => [3,3]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => [3,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => [3,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1] => [3,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1] => [3,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => [3,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => [3,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,1] => [3,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,1] => [3,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => [3,3]
=> 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? => ?
=> ? = 4
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? => ?
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? => ?
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 3
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000069
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ([(0,3),(0,7),(4,6),(5,2),(6,5),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(3,5),(4,3),(5,2),(6,1),(7,4),(7,6)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> ([(0,7),(1,6),(2,3),(3,5),(4,7),(5,6),(6,4)],8)
=> ([(0,3),(0,6),(4,5),(5,2),(6,7),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ([(0,7),(4,5),(5,3),(6,2),(6,4),(7,1),(7,6)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ([(0,6),(3,4),(4,1),(5,2),(6,7),(7,3),(7,5)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,2),(5,3),(6,4),(7,1),(7,5)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,2),(5,3),(6,4),(7,1),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
=> ([(0,3),(0,6),(4,7),(5,2),(6,4),(7,1),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ([(0,7),(4,6),(5,3),(6,2),(6,5),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ([(0,5),(4,3),(5,6),(6,2),(6,7),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,2),(5,3),(6,4),(7,1),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> ([(0,7),(1,5),(2,5),(3,4),(4,7),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(5,4),(6,2),(6,3),(7,1),(7,5)],8)
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,2),(5,3),(6,4),(7,1),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,5),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,3),(5,2),(6,1),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(4,1),(5,6),(6,2),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(3,5),(4,3),(5,2),(6,1),(7,4),(7,6)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,5),(2,6),(3,4),(4,7),(5,7),(7,6)],8)
=> ([(0,4),(0,7),(5,3),(6,1),(6,2),(7,5),(7,6)],8)
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ([(0,7),(4,2),(5,3),(6,1),(6,5),(7,4),(7,6)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> ([(0,4),(1,3),(2,5),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,3),(5,2),(6,1),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,5),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(4,3),(5,2),(6,1),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(4,1),(5,6),(6,2),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ([(0,7),(4,2),(5,3),(6,1),(6,5),(7,4),(7,6)],8)
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ([(0,3),(0,6),(4,5),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> ([(0,6),(1,6),(2,7),(4,5),(5,7),(6,4),(7,3)],8)
=> ([(0,7),(4,5),(5,6),(6,2),(6,3),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,7),(5,7),(6,4)],8)
=> ([(0,5),(0,6),(4,1),(5,4),(6,7),(7,2),(7,3)],8)
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],[[.,.],[.,.]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,4)],8)
=> ([(0,5),(0,7),(5,6),(6,3),(6,4),(7,1),(7,2)],8)
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,7),(5,7),(6,4)],8)
=> ([(0,5),(0,6),(4,1),(5,4),(6,7),(7,2),(7,3)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ([(0,5),(0,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ([(0,5),(0,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,5),(2,6),(3,4),(4,7),(5,7),(7,6)],8)
=> ([(0,4),(0,7),(5,3),(6,1),(6,2),(7,5),(7,6)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> ([(0,6),(1,6),(2,4),(4,5),(5,7),(6,7),(7,3)],8)
=> ([(0,7),(4,5),(5,3),(6,1),(6,2),(7,4),(7,6)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ([(0,6),(1,6),(2,5),(3,5),(5,7),(6,7),(7,4)],8)
=> ([(0,7),(5,3),(5,4),(6,1),(6,2),(7,5),(7,6)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> ([(0,7),(1,5),(2,5),(3,4),(4,7),(5,6),(7,6)],8)
=> ([(0,6),(0,7),(5,4),(6,2),(6,3),(7,1),(7,5)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[[[.,.],[.,.]],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,5),(2,6),(3,4),(4,7),(5,7),(7,6)],8)
=> ([(0,4),(0,7),(5,3),(6,1),(6,2),(7,5),(7,6)],8)
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> ([(0,6),(1,6),(2,4),(4,5),(5,7),(6,7),(7,3)],8)
=> ([(0,7),(4,5),(5,3),(6,1),(6,2),(7,4),(7,6)],8)
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,4)],8)
=> ([(0,5),(0,7),(5,6),(6,3),(6,4),(7,1),(7,2)],8)
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,6),(5,7),(6,5)],8)
=> ([(0,5),(0,6),(4,1),(5,3),(6,7),(7,2),(7,4)],8)
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ([(0,4),(0,7),(5,3),(6,2),(6,5),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> ([(0,6),(1,3),(2,5),(3,7),(5,6),(6,7),(7,4)],8)
=> ([(0,7),(4,2),(5,3),(6,1),(6,5),(7,4),(7,6)],8)
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(4,1),(5,6),(6,2),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ([(0,4),(0,7),(5,3),(6,2),(6,5),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> ([(0,6),(1,4),(2,3),(3,7),(4,7),(5,6),(7,5)],8)
=> ([(0,3),(0,6),(4,2),(5,1),(6,7),(7,4),(7,5)],8)
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> ([(0,6),(1,6),(2,7),(3,5),(4,7),(6,4),(7,5)],8)
=> ([(0,4),(0,7),(5,6),(6,2),(6,3),(7,1),(7,5)],8)
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> ([(0,6),(1,7),(2,5),(3,5),(4,7),(5,6),(6,4)],8)
=> ([(0,4),(0,5),(5,7),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ([(0,3),(0,7),(4,6),(5,2),(6,5),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ([(0,4),(0,7),(5,3),(6,2),(6,5),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ([(0,4),(0,7),(5,3),(6,2),(6,5),(7,1),(7,6)],8)
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ([(0,3),(0,7),(4,6),(5,2),(6,5),(7,1),(7,4)],8)
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],[.,[.,.]]],.],[[.,.],.]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,6),(5,7),(6,5)],8)
=> ([(0,5),(0,6),(4,1),(5,3),(6,7),(7,2),(7,4)],8)
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,6),(5,7),(6,5)],8)
=> ([(0,5),(0,6),(4,1),(5,3),(6,7),(7,2),(7,4)],8)
=> ? = 3
Description
The number of maximal elements of a poset.
The following 107 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000071The number of maximal chains in a poset. St000527The width of the poset. St001732The number of peaks visible from the left. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000068The number of minimal elements in a poset. St000010The length of the partition. St000257The number of distinct parts of a partition that occur at least twice. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000632The jump number of the poset. St001840The number of descents of a set partition. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001304The number of maximally independent sets of vertices of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001083The number of boxed occurrences of 132 in a permutation. St000157The number of descents of a standard tableau. St000647The number of big descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000159The number of distinct parts of the integer partition. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St001092The number of distinct even parts of a partition. St000779The tier of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000256The number of parts from which one can substract 2 and still get an integer partition. St000312The number of leaves in a graph. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000710The number of big deficiencies of a permutation. St000646The number of big ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000360The number of occurrences of the pattern 32-1. St000092The number of outer peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000409The number of pitchforks in a binary tree. St000357The number of occurrences of the pattern 12-3. St000636The hull number of a graph. St001307The number of induced stars on four vertices in a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St001883The mutual visibility number of a graph. St001597The Frobenius rank of a skew partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000834The number of right outer peaks of a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001330The hat guessing number of a graph. St000742The number of big ascents of a permutation after prepending zero. St000245The number of ascents of a permutation. St000640The rank of the largest boolean interval in a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001729The number of visible descents of a permutation. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000252The number of nodes of degree 3 of a binary tree. St000021The number of descents of a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000991The number of right-to-left minima of a permutation. St001427The number of descents of a signed permutation. St001487The number of inner corners of a skew partition. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001964The interval resolution global dimension of a poset. St001960The number of descents of a permutation minus one if its first entry is not one. St000624The normalized sum of the minimal distances to a greater element. St000805The number of peaks of the associated bargraph. St000807The sum of the heights of the valleys of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001469The holeyness of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000768The number of peaks in an integer composition.
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