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Your data matches 218 different statistics following compositions of up to 3 maps.
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Matching statistic: St000201
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 1
[.,[.,.]]
=> 1
[[.,.],.]
=> 1
[.,[.,[.,.]]]
=> 1
[.,[[.,.],.]]
=> 1
[[.,.],[.,.]]
=> 2
[[.,[.,.]],.]
=> 1
[[[.,.],.],.]
=> 1
[.,[.,[.,[.,.]]]]
=> 1
[.,[.,[[.,.],.]]]
=> 1
[.,[[.,.],[.,.]]]
=> 2
[.,[[.,[.,.]],.]]
=> 1
[.,[[[.,.],.],.]]
=> 1
[[.,.],[.,[.,.]]]
=> 2
[[.,.],[[.,.],.]]
=> 2
[[.,[.,.]],[.,.]]
=> 2
[[[.,.],.],[.,.]]
=> 2
[[.,[.,[.,.]]],.]
=> 1
[[.,[[.,.],.]],.]
=> 1
[[[.,.],[.,.]],.]
=> 2
[[[.,[.,.]],.],.]
=> 1
[[[[.,.],.],.],.]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> 1
[.,[[[[.,.],.],.],.]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> 2
[[.,.],[[[.,.],.],.]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> 2
[[[.,.],.],[[.,.],.]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> 3
[[[.,[.,.]],.],[.,.]]
=> 2
[[[[.,.],.],.],[.,.]]
=> 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 $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000291
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => 1 => 1 => 0 = 1 - 1
[[.,.],.]
=> [1,2] => 0 => 0 => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 11 => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => 10 => 01 => 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => 01 => 10 => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 01 => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 00 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 111 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 110 => 011 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 101 => 101 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 110 => 011 => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 100 => 001 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 110 => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 010 => 010 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 011 => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 100 => 001 => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 001 => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 1111 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1110 => 0111 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1101 => 1011 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1110 => 0111 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1100 => 0011 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1011 => 1101 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1010 => 0101 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1101 => 1011 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1001 => 1001 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1110 => 0111 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1100 => 0011 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1010 => 0101 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1100 => 0011 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1000 => 0001 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 1110 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0110 => 0110 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0101 => 1010 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0110 => 0110 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0100 => 0010 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1101 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1010 => 0101 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 1100 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0010 => 0100 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 1011 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1001 => 1001 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 1010 => 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1001 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 0111 => 0 = 1 - 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => ? => ? => ? = 3 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => ? => ? => ? = 3 - 1
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => ? => ? => ? = 3 - 1
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ? => ? => ? = 3 - 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> [1,3,6,7,5,8,4,2] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,4,5,3,7,8,6,2] => ? => ? => ? = 3 - 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ? => ? => ? = 4 - 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ? => ? => ? = 4 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ? => ? => ? = 3 - 1
[[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [2,3,1,5,7,8,6,4] => ? => ? => ? = 3 - 1
[[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [2,3,1,5,6,8,7,4] => ? => ? => ? = 3 - 1
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ? => ? => ? = 3 - 1
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ? => ? => ? = 3 - 1
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [2,1,3,4,6,8,7,5] => ? => ? => ? = 3 - 1
[[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ? => ? => ? = 4 - 1
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ? => ? => ? = 3 - 1
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ? => ? => ? = 4 - 1
[[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [3,4,2,6,7,5,1,8] => ? => ? => ? = 2 - 1
[[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [3,5,6,4,2,7,1,8] => ? => ? => ? = 2 - 1
[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,2,7,1,8] => ? => ? => ? = 2 - 1
[[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => ? => ? => ? = 3 - 1
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,4,3,6,9,8,7,5] => ? => ? => ? = 4 - 1
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,5,4,7,9,8,6,1] => ? => ? => ? = 4 - 1
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,4,7,10,9,8,6] => ? => ? => ? = 5 - 1
[[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,3,6,9,8,7,5] => ? => ? => ? = 4 - 1
[.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,4,7,9,8,6,1] => ? => ? => ? = 4 - 1
[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => ? => ? => ? = 5 - 1
[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,6,4,8,11,10,9,7] => ? => ? => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,4,7,10,9,8,6] => ? => ? => ? = 4 - 1
[[[.,.],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,4,2,6,9,8,7,5] => ? => ? => ? = 4 - 1
[[[.,.],[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [1,3,2,5,8,7,6,9,4] => ? => ? => ? = 4 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,5,3,7,9,8,6,1] => ? => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[[[.,.],[.,.]],.]]]
=> [2,4,3,6,8,7,9,5,1] => ? => ? => ? = 4 - 1
[.,[[[[[.,.],.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,4,6,5,8,10,9,7,1] => ? => ? => ? = 4 - 1
[.,[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,3,6,7,5,9,11,10,8,1] => ? => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,3,5,8,9,7,10,6,4,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,3,6,7,5,9,10,8,4,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,3,6,8,9,7,5,10,4,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [1,3,7,8,6,9,5,10,4,2] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [1,4,5,3,7,9,10,8,6,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [1,4,5,3,8,9,7,10,6,2] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [1,4,6,7,5,3,9,10,8,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [1,5,6,4,7,3,9,10,8,2] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [1,4,6,8,9,7,5,3,10,2] => ? => ? => ? = 4 - 1
[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [1,4,7,8,6,9,5,3,10,2] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [1,5,6,4,8,9,7,3,10,2] => ? => ? => ? = 3 - 1
[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [1,5,7,8,6,4,9,3,10,2] => ? => ? => ? = 3 - 1
[[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [2,3,1,5,7,9,10,8,6,4] => ? => ? => ? = 4 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => 1 => 0 = 1 - 1
[[.,.],.]
=> [1,2] => 0 => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => 10 => 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => 01 => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => 10 => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 110 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 110 => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 100 => 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 010 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 100 => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1110 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1101 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1110 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1100 => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1011 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1010 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1101 => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1001 => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1110 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1100 => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1010 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1100 => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1000 => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0110 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0101 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0110 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0100 => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1010 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0010 => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1001 => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 1 = 2 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 0 = 1 - 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => ? => ? = 3 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => ? => ? = 3 - 1
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => ? => ? = 3 - 1
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ? => ? = 3 - 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> [1,3,6,7,5,8,4,2] => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,4,5,3,7,8,6,2] => ? => ? = 3 - 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ? => ? = 4 - 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ? => ? = 4 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ? => ? = 3 - 1
[[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [2,3,1,5,7,8,6,4] => ? => ? = 3 - 1
[[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [2,3,1,5,6,8,7,4] => ? => ? = 3 - 1
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ? => ? = 3 - 1
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ? => ? = 3 - 1
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [2,1,3,4,6,8,7,5] => ? => ? = 3 - 1
[[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ? => ? = 4 - 1
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ? => ? = 3 - 1
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ? => ? = 4 - 1
[[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [3,4,2,6,7,5,1,8] => ? => ? = 2 - 1
[[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [3,5,6,4,2,7,1,8] => ? => ? = 2 - 1
[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,2,7,1,8] => ? => ? = 2 - 1
[[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => ? => ? = 3 - 1
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,4,3,6,9,8,7,5] => ? => ? = 4 - 1
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,5,4,7,9,8,6,1] => ? => ? = 4 - 1
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,4,7,10,9,8,6] => ? => ? = 5 - 1
[[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,3,6,9,8,7,5] => ? => ? = 4 - 1
[.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,4,7,9,8,6,1] => ? => ? = 4 - 1
[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => ? => ? = 5 - 1
[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,6,4,8,11,10,9,7] => ? => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,4,7,10,9,8,6] => ? => ? = 4 - 1
[[[.,.],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,4,2,6,9,8,7,5] => ? => ? = 4 - 1
[[[.,.],[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [1,3,2,5,8,7,6,9,4] => ? => ? = 4 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,5,3,7,9,8,6,1] => ? => ? = 4 - 1
[.,[[[.,.],[.,.]],[[[.,.],[.,.]],.]]]
=> [2,4,3,6,8,7,9,5,1] => ? => ? = 4 - 1
[.,[[[[[.,.],.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,4,6,5,8,10,9,7,1] => ? => ? = 4 - 1
[.,[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,3,6,7,5,9,11,10,8,1] => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,3,5,8,9,7,10,6,4,2] => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,3,6,7,5,9,10,8,4,2] => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,3,6,8,9,7,5,10,4,2] => ? => ? = 4 - 1
[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [1,3,7,8,6,9,5,10,4,2] => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [1,4,5,3,7,9,10,8,6,2] => ? => ? = 4 - 1
[[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [1,4,5,3,8,9,7,10,6,2] => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [1,4,6,7,5,3,9,10,8,2] => ? => ? = 4 - 1
[[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [1,5,6,4,7,3,9,10,8,2] => ? => ? = 3 - 1
[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [1,4,6,8,9,7,5,3,10,2] => ? => ? = 4 - 1
[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [1,4,7,8,6,9,5,3,10,2] => ? => ? = 3 - 1
[[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [1,5,6,4,8,9,7,3,10,2] => ? => ? = 3 - 1
[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [1,5,7,8,6,4,9,3,10,2] => ? => ? = 3 - 1
[[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [2,3,1,5,7,9,10,8,6,4] => ? => ? = 4 - 1
Description
The number of ascents of a binary word.
Matching statistic: St001712
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [.,.]
=> [1] => [[1]]
=> 0 = 1 - 1
[.,[.,.]]
=> [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 0 = 1 - 1
[[.,.],.]
=> [[.,.],.]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => [[1,2],[3]]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,3,2] => [[1,2],[3]]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> [4,5,3,2,7,6,8,1] => ?
=> ? = 3 - 1
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [3,2,4,7,6,8,5,1] => ?
=> ? = 3 - 1
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> [2,4,7,6,8,5,3,1] => ?
=> ? = 2 - 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,2,7,1,8] => ?
=> ? = 4 - 1
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> [4,3,2,7,6,5,1,8] => ?
=> ? = 3 - 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [[.,[[[[.,[.,.]],[.,.]],.],.]],.]
=> [3,2,5,4,6,7,1,8] => ?
=> ? = 4 - 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [[.,[[[[.,[.,.]],.],[.,.]],.]],.]
=> [3,2,4,6,5,7,1,8] => ?
=> ? = 4 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [[.,[[[[.,[.,.]],.],.],[.,.]]],.]
=> [3,2,4,5,7,6,1,8] => ?
=> ? = 3 - 1
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [[.,[[[[.,.],[[.,.],.]],.],.]],.]
=> [2,4,5,3,6,7,1,8] => ?
=> ? = 3 - 1
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> [[.,[[[.,[.,.]],.],[.,.]]],[.,.]]
=> [3,2,4,6,5,1,8,7] => ?
=> ? = 3 - 1
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> [[.,[[[[.,[.,.]],.],.],.]],[.,.]]
=> [3,2,4,5,6,1,8,7] => ?
=> ? = 3 - 1
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> [[.,[[.,.],[[.,[.,.]],.]]],[.,.]]
=> [2,5,4,6,3,1,8,7] => ?
=> ? = 3 - 1
[[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [[.,[[[.,[.,.]],.],.]],[[.,.],.]]
=> [3,2,4,5,1,7,8,6] => ?
=> ? = 3 - 1
[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
=> [[[.,[.,[[.,[.,.]],.]]],[.,.]],.]
=> [4,3,5,2,1,7,6,8] => ?
=> ? = 4 - 1
[[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> [[[.,[[.,[.,[.,.]]],.]],[.,.]],.]
=> [4,3,2,5,1,7,6,8] => ?
=> ? = 4 - 1
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [[[.,[.,[[.,.],.]]],[.,[.,.]]],.]
=> [3,4,2,1,7,6,5,8] => ?
=> ? = 3 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [[[.,[[.,.],[.,.]]],[.,[.,.]]],.]
=> [2,4,3,1,7,6,5,8] => ?
=> ? = 3 - 1
[[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [[[[.,[.,[[.,.],.]]],[.,.]],.],.]
=> [3,4,2,1,6,5,7,8] => ?
=> ? = 3 - 1
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [[[[.,[.,[[.,.],.]]],.],[.,.]],.]
=> [3,4,2,1,5,7,6,8] => ?
=> ? = 3 - 1
[[[[.,[.,.]],.],.],[.,[[.,.],.]]]
=> [[[[.,[.,[[.,.],.]]],.],.],[.,.]]
=> [3,4,2,1,5,6,8,7] => ?
=> ? = 2 - 1
[[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [3,2,1,5,4,6,8,7] => ?
=> ? = 3 - 1
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [[[[.,[[.,.],.]],[.,.]],.],[.,.]]
=> [2,3,1,5,4,6,8,7] => ?
=> ? = 3 - 1
[[[[.,.],[.,[.,.]]],.],[[.,.],.]]
=> [[[[.,[[.,.],.]],.],[.,[.,.]]],.]
=> [2,3,1,4,7,6,5,8] => ?
=> ? = 3 - 1
[[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [3,2,1,4,6,5,8,7] => ?
=> ? = 3 - 1
[[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],.],.]
=> [2,1,5,4,6,3,7,8] => ?
=> ? = 4 - 1
[[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [[[[.,[.,.]],[[[.,.],.],.]],.],.]
=> [2,1,4,5,6,3,7,8] => ?
=> ? = 3 - 1
[[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [[[[.,[.,.]],[[.,.],.]],[.,.]],.]
=> [2,1,4,5,3,7,6,8] => ?
=> ? = 4 - 1
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 4 - 1
[[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [[[[.,[.,.]],[.,.]],[[.,.],.]],.]
=> [2,1,4,3,6,7,5,8] => ?
=> ? = 4 - 1
[[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 3 - 1
[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,5,4,6,3,8,7,2] => ?
=> ? = 3 - 1
[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ?
=> ? = 2 - 1
[[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 2 - 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1 - 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => [[1,2,3,4,5,6,7,8,9],[10]]
=> ? = 1 - 1
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [[[[.,[[.,[.,[.,.]]],.]],[.,.]],.],.]
=> [4,3,2,5,1,7,6,8,9] => ?
=> ? = 4 - 1
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],.],.]]
=> [4,3,5,2,7,6,8,9,1] => ?
=> ? = 4 - 1
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [[[[.,[[.,[.,[.,.]]],.]],[.,.]],[.,.]],.]
=> [4,3,2,5,1,7,6,9,8,10] => ?
=> ? = 5 - 1
[[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [[[.,[[.,[.,[.,.]]],.]],[.,.]],[.,.]]
=> [4,3,2,5,1,7,6,9,8] => ?
=> ? = 4 - 1
[.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]]
=> [4,3,5,2,7,6,9,8,1] => ?
=> ? = 4 - 1
[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [4,3,5,2,7,6,9,8,10,1] => [[1,3,5,7,9],[2,6,8],[4],[10]]
=> ? = 5 - 1
[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [[[[.,[[.,[.,[.,.]]],.]],[[.,.],.]],[.,.]],.]
=> [4,3,2,5,1,7,8,6,10,9,11] => ?
=> ? = 5 - 1
[[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [[[[[.,[[.,[.,[.,.]]],.]],[.,.]],.],.],.]
=> [4,3,2,5,1,7,6,8,9,10] => ?
=> ? = 4 - 1
[[[.,.],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [[[.,[[.,[.,[.,.]]],.]],[[.,.],.]],.]
=> [4,3,2,5,1,7,8,6,9] => ?
=> ? = 4 - 1
[[[.,.],[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [[[.,[[[.,.],[.,[.,.]]],.]],[.,.]],.]
=> [2,5,4,3,6,1,8,7,9] => ?
=> ? = 4 - 1
[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,.],.]],.]]
=> [4,3,5,2,7,8,6,9,1] => ?
=> ? = 4 - 1
[.,[[[.,.],[.,.]],[[[.,.],[.,.]],.]]]
=> [.,[[[.,[[[.,.],[.,.]],.]],[.,.]],.]]
=> [3,5,4,6,2,8,7,9,1] => ?
=> ? = 4 - 1
[.,[[[[[.,.],.],.],[.,.]],[[.,.],[.,.]]]]
=> [.,[[[[[.,[[.,[.,.]],.]],[.,.]],.],.],.]]
=> [4,3,5,2,7,6,8,9,10,1] => ?
=> ? = 4 - 1
[.,[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,.]]]]
=> [.,[[[[.,[[.,[.,.]],.]],[[.,.],.]],[.,.]],.]]
=> [4,3,5,2,7,8,6,10,9,11,1] => ?
=> ? = 5 - 1
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,9,8,7,6,5,4,3,2] => [[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: St000386
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [7,8,6,4,2,3,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 4 - 1
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [7,8,5,2,3,4,6,1] => ?
=> ?
=> ? = 3 - 1
[[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [7,8,6,4,3,5,1,2] => ?
=> ?
=> ? = 3 - 1
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [8,6,7,4,3,5,1,2] => ?
=> ?
=> ? = 4 - 1
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [8,7,5,4,3,6,1,2] => ?
=> ?
=> ? = 3 - 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [8,6,3,4,5,7,1,2] => ?
=> ?
=> ? = 4 - 1
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,4,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [7,8,6,4,5,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,2,1,3] => ?
=> ?
=> ? = 4 - 1
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [8,7,5,4,6,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> [8,6,4,5,7,2,1,3] => ?
=> ?
=> ? = 4 - 1
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> [8,5,4,6,7,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> [7,5,6,4,8,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,2,1,3] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [8,6,7,4,2,3,1,5] => ?
=> ?
=> ? = 4 - 1
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [7,6,8,3,2,4,1,5] => ?
=> ?
=> ? = 2 - 1
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [7,8,6,4,3,1,2,5] => ?
=> ?
=> ? = 3 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,1,2,5] => ?
=> ?
=> ? = 3 - 1
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [7,8,6,4,1,2,3,5] => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,1,2,3,5] => ?
=> ?
=> ? = 4 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,1,2,3,5] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [7,8,5,4,1,2,3,6] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [7,8,4,5,1,2,3,6] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [8,7,4,3,1,2,5,6] => ?
=> ?
=> ? = 3 - 1
[[[.,[.,.]],[[.,.],[.,.]]],[.,.]]
=> [8,6,4,5,2,1,3,7] => ?
=> ?
=> ? = 4 - 1
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [8,6,4,5,1,2,3,7] => [[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,1,2,3,7] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],[.,.]],[.,[.,.]]],[.,.]]
=> [8,6,5,3,1,2,4,7] => ?
=> ?
=> ? = 4 - 1
[[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]
=> [8,5,3,2,4,1,6,7] => ?
=> ?
=> ? = 3 - 1
[[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [7,5,6,3,2,4,1,8] => ?
=> ?
=> ? = 3 - 1
[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [6,4,3,5,2,7,1,8] => ?
=> ?
=> ? = 2 - 1
[[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [6,7,4,5,1,2,3,8] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [7,5,4,6,1,2,3,8] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [4,5,6,7,1,2,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> [6,7,5,3,1,2,4,8] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [7,5,6,3,1,2,4,8] => ?
=> ?
=> ? = 4 - 1
[[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [7,4,1,2,3,5,6,8] => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => ?
=> ?
=> ? = 3 - 1
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ?
=> ?
=> ? = 3 - 1
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [3,5,4,7,6,2,8,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [4,6,5,3,7,2,8,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ?
=> ?
=> ? = 4 - 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ?
=> ?
=> ? = 4 - 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ?
=> ?
=> ? = 4 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ?
=> ?
=> ? = 3 - 1
[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [3,2,1,7,8,6,5,4] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [3,2,1,6,8,7,5,4] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [3,2,1,6,7,8,5,4] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [3,2,1,5,8,7,6,4] => [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [3,2,1,5,7,8,6,4] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[.,[.,[.,.]]],[[[.,.],.],[.,.]]]
=> [3,2,1,5,6,8,7,4] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [2,3,1,5,6,8,7,4] => ?
=> ?
=> ? = 3 - 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [3,4,2,1,7,8,6,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [3,2,4,1,6,8,7,5] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ?
=> ?
=> ? = 3 - 1
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ?
=> ?
=> ? = 3 - 1
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [2,1,3,4,6,8,7,5] => ?
=> ?
=> ? = 3 - 1
[[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [3,4,2,5,1,7,8,6] => [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ?
=> ?
=> ? = 4 - 1
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ?
=> ?
=> ? = 3 - 1
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 4 - 1
[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]
=> [3,2,5,4,1,6,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [3,2,1,4,5,6,8,7] => [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [5,7,6,4,3,2,1,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [6,5,7,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [5,6,7,4,3,2,1,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [4,7,6,5,3,2,1,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [6,5,4,7,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [.,.]
=> ? = 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> 1
[[.,.],[.,.]]
=> [1,3,2] => [.,[[.,.],.]]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => ?
=> ? = 3
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => ?
=> ? = 3
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => ?
=> ? = 3
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ?
=> ? = 3
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [3,5,4,7,6,2,8,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [4,6,5,3,7,2,8,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ?
=> ? = 4
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ?
=> ? = 4
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ?
=> ? = 4
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ?
=> ? = 3
[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [3,2,1,7,8,6,5,4] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [3,2,1,6,8,7,5,4] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 3
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [3,2,1,6,7,8,5,4] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [3,2,1,5,8,7,6,4] => [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> ? = 3
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [3,2,1,5,7,8,6,4] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[[.,[.,[.,.]]],[[[.,.],.],[.,.]]]
=> [3,2,1,5,6,8,7,4] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [2,3,1,5,6,8,7,4] => ?
=> ? = 3
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 2
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [3,4,2,1,7,8,6,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [3,2,4,1,6,8,7,5] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ?
=> ? = 3
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ?
=> ? = 3
[[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [2,1,3,4,6,8,7,5] => ?
=> ? = 3
[[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [3,4,2,5,1,7,8,6] => [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> ? = 2
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> ? = 2
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 2
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ?
=> ? = 4
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ?
=> ? = 3
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 4
[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]
=> [3,2,5,4,1,6,8,7] => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [3,2,1,4,5,6,8,7] => [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ? = 2
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [5,7,6,4,3,2,1,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [6,5,7,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [5,6,7,4,3,2,1,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [4,7,6,5,3,2,1,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 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: St000069
Values
[.,.]
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> ([(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
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> ([(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
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(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
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> ([(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
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> ([(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
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> ([(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
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> ([(0,6),(1,6),(2,3),(3,7),(5,7),(6,5),(7,4)],8)
=> ([(0,7),(4,3),(5,6),(6,1),(6,2),(7,4),(7,5)],8)
=> ? = 3
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ([(0,7),(5,2),(5,6),(6,3),(6,4),(7,1),(7,5)],8)
=> ? = 4
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> ([(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
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 3
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 2
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 2
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> ([(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
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 2
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> ([(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
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 2
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(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
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(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
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(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
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(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
[[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> ([(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
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ([(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
[[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(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
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(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
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> ([(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
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(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
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> ([(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
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> ([(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
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> ([(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
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> ([(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
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> ([(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
[[.,.],[[.,[[.,.],[[.,.],.]]],.]]
=> ([(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
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> ([(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
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> ([(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
[[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> ([(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
[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> ([(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
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ([(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
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> ([(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
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> ([(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
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> ([(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
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(5,7),(6,4)],8)
=> ([(0,5),(0,6),(4,7),(5,3),(6,4),(7,1),(7,2)],8)
=> ? = 3
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> ([(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
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> ([(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
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> ([(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
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ([(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
Description
The number of maximal elements of a poset.
Matching statistic: St001732
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => [2,4,8,7,6,3,5,1] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [7,8,6,4,2,3,5,1] => [2,4,7,8,6,3,5,1] => ?
=> ? = 3
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [8,6,7,4,2,3,5,1] => [2,4,6,8,7,3,5,1] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [7,6,8,4,2,3,5,1] => [2,4,7,6,8,3,5,1] => [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [6,7,8,4,2,3,5,1] => [2,4,6,7,8,3,5,1] => [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 3
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [7,8,5,2,3,4,6,1] => [2,3,5,7,8,4,6,1] => [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [7,8,4,2,3,5,6,1] => [2,4,3,7,8,5,6,1] => [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [8,6,4,2,3,5,7,1] => [2,4,6,8,3,5,7,1] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [8,5,3,4,2,6,7,1] => [3,5,4,8,2,6,7,1] => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 3
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [7,5,3,4,6,2,8,1] => [3,5,7,4,6,2,8,1] => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 3
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [6,4,5,3,7,2,8,1] => [4,6,5,3,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [6,4,3,5,7,2,8,1] => [4,6,3,5,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [6,5,7,3,2,4,8,1] => [3,6,5,7,2,4,8,1] => ?
=> ? = 2
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [7,4,3,5,2,6,8,1] => [4,3,7,5,2,6,8,1] => ?
=> ? = 2
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [7,5,3,2,4,6,8,1] => [3,5,7,2,4,6,8,1] => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 3
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [2,3,5,4,6,7,8,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [2,4,3,5,6,7,8,1] => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,4,1,2] => [1,3,6,8,5,7,4,2] => ?
=> ? = 4
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [8,6,3,4,5,7,1,2] => [1,3,4,6,8,5,7,2] => ?
=> ? = 4
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [8,5,3,4,6,7,1,2] => [1,3,5,4,8,6,7,2] => ?
=> ? = 4
[[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [7,8,5,6,4,2,1,3] => [2,5,7,8,6,4,1,3] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,4,2,1,3] => [2,6,8,5,7,4,1,3] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,2,1,3] => [2,5,6,7,8,4,1,3] => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [7,8,6,4,5,2,1,3] => [2,4,7,8,6,5,1,3] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [8,6,7,4,5,2,1,3] => [2,4,6,8,7,5,1,3] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,2,1,3] => [2,4,6,7,8,5,1,3] => [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 3
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [8,7,5,4,6,2,1,3] => [2,5,8,7,4,6,1,3] => [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> [7,8,4,5,6,2,1,3] => [2,4,5,7,8,6,1,3] => ?
=> ? = 3
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> [8,6,4,5,7,2,1,3] => [2,4,6,8,5,7,1,3] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> [8,5,4,6,7,2,1,3] => [2,5,4,8,6,7,1,3] => ?
=> ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> [8,4,5,6,7,2,1,3] => [2,4,5,6,8,7,1,3] => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> [7,5,6,4,8,2,1,3] => [2,5,7,6,4,8,1,3] => ?
=> ? = 3
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [6,5,7,4,8,2,1,3] => [2,6,5,7,4,8,1,3] => ?
=> ? = 2
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> [6,7,4,5,8,2,1,3] => [2,4,6,7,5,8,1,3] => [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 3
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => [2,5,7,4,6,8,1,3] => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,2,1,3] => [2,4,5,6,7,8,1,3] => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [7,8,6,5,3,2,1,4] => [3,7,8,6,5,2,1,4] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [8,6,7,5,3,2,1,4] => [3,6,8,7,5,2,1,4] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 3
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [6,7,8,5,3,2,1,4] => [3,6,7,8,5,2,1,4] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,3,2,1,4] => [3,5,8,7,6,2,1,4] => ?
=> ? = 3
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,3,2,1,4] => [3,5,7,8,6,2,1,4] => [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[[.,[.,[.,.]]],[[[.,.],.],[.,.]]]
=> [8,5,6,7,3,2,1,4] => [3,5,6,8,7,2,1,4] => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3
[[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,2,3,1,4] => [2,3,5,7,8,6,1,4] => ?
=> ? = 3
[[.,[[.,.],.]],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,2,3,1,4] => [2,3,6,8,5,7,1,4] => ?
=> ? = 3
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000071
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> ([(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
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> ([(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
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> ([(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
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> ([(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
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> ([(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
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> ([(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
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> ([(0,6),(1,6),(2,3),(3,7),(5,7),(6,5),(7,4)],8)
=> ([(0,7),(4,3),(5,6),(6,1),(6,2),(7,4),(7,5)],8)
=> ? = 3
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ([(0,7),(5,2),(5,6),(6,3),(6,4),(7,1),(7,5)],8)
=> ? = 4
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> ([(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
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 3
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 2
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 2
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> ([(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
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 2
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> ([(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
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 2
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 2
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(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
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(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
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(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
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(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
[[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> ([(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
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ([(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
[[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(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
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(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
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> ([(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
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(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
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> ([(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
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> ([(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
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> ([(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
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> ([(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
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> ([(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
[[.,.],[[.,[[.,.],[[.,.],.]]],.]]
=> ([(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
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> ([(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
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> ([(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
[[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> ([(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
[[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> ([(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
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ([(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
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> ([(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
[[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> ([(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
[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> ([(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
[[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ([(0,6),(0,7),(4,5),(5,3),(6,1),(7,2),(7,4)],8)
=> ? = 3
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(5,7),(6,4)],8)
=> ([(0,5),(0,6),(4,7),(5,3),(6,4),(7,1),(7,2)],8)
=> ? = 3
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> ([(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
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> ([(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
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> ([(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
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ([(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
Description
The number of maximal chains in a poset.
The following 208 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000527The width of the poset. St000068The number of minimal elements in a poset. St000010The length of the partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001083The number of boxed occurrences of 132 in a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. 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. St000820The number of compositions obtained by rotating the composition. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St000387The matching number of a graph. St000778The metric dimension of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001777The number of weak descents in an integer composition. St001840The number of descents of a set partition. St000299The number of nonisomorphic vertex-induced subtrees. St000553The number of blocks of a graph. St000822The Hadwiger number of the graph. St001642The Prague dimension of a graph. St001644The dimension of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001812The biclique partition number of a graph. St000257The number of distinct parts of a partition that occur at least twice. 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. St001176The size of a partition minus its first part. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000659The number of rises of length at least 2 of a Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000741The Colin de Verdière graph invariant. St000710The number of big deficiencies of a permutation. St000251The number of nonsingleton blocks of a set partition. St000360The number of occurrences of the pattern 32-1. St000523The number of 2-protected nodes of a rooted tree. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000409The number of pitchforks in a binary tree. St000023The number of inner peaks of a permutation. St000646The number of big ascents of a permutation. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000919The number of maximal left branches of a binary tree. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001580The acyclic chromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001792The arboricity of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001597The Frobenius rank of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St001060The distinguishing index of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000834The number of right outer peaks of a permutation. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000994The number of cycle peaks and the number of cycle valleys 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. St000703The number of deficiencies of a permutation. St000640The rank of the largest boolean interval in a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001729The number of visible descents of a permutation. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. 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. St000252The number of nodes of degree 3 of a binary tree. St000325The width of the tree associated to a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000470The number of runs in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset. St000807The sum of the heights of the valleys of the associated bargraph. St000264The girth of a graph, which is not a tree. St000381The largest part of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000781The number of proper colouring schemes of a Ferrers diagram. St000805The number of peaks of the associated bargraph. St000808The number of up steps of the associated bargraph. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St001673The degree of asymmetry of an integer composition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian.
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