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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000201
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Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> 1
[2,1] => [1] => [.,.]
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> 1
[2,3,1] => [2,1] => [[.,.],.]
=> 1
[3,1,2] => [1,2] => [.,[.,.]]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
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
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(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => => ? = 1 - 1
[2,1] => [1] => => ? = 1 - 1
[1,2,3] => [1,2] => 0 => 0 = 1 - 1
[1,3,2] => [1,2] => 0 => 0 = 1 - 1
[2,1,3] => [2,1] => 1 => 0 = 1 - 1
[2,3,1] => [2,1] => 1 => 0 = 1 - 1
[3,1,2] => [1,2] => 0 => 0 = 1 - 1
[3,2,1] => [2,1] => 1 => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => 00 => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => 01 => 0 = 1 - 1
[1,3,4,2] => [1,3,2] => 01 => 0 = 1 - 1
[1,4,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => 01 => 0 = 1 - 1
[2,1,3,4] => [2,1,3] => 10 => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => 10 => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => 01 => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => 01 => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => 01 => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => 10 => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => 10 => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => 11 => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => 11 => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => 10 => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => 01 => 0 = 1 - 1
[4,2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => 01 => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => 10 => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => 001 => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => 001 => 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => 001 => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => 011 => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[1,5,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,5,2,4,3] => [1,2,4,3] => 001 => 0 = 1 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
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(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => => => ? = 1 - 1
[2,1] => [1] => => => ? = 1 - 1
[1,2,3] => [1,2] => 0 => 0 => 0 = 1 - 1
[1,3,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1,3] => [2,1] => 1 => 1 => 0 = 1 - 1
[2,3,1] => [2,1] => 1 => 1 => 0 = 1 - 1
[3,1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[3,2,1] => [2,1] => 1 => 1 => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => 01 => 10 => 0 = 1 - 1
[1,3,4,2] => [1,3,2] => 01 => 10 => 0 = 1 - 1
[1,4,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => 01 => 10 => 0 = 1 - 1
[2,1,3,4] => [2,1,3] => 10 => 01 => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => 10 => 01 => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => 01 => 10 => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => 01 => 10 => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => 10 => 01 => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => 01 => 10 => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => 10 => 01 => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => 10 => 01 => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => 10 => 01 => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => 01 => 10 => 0 = 1 - 1
[4,2,1,3] => [2,1,3] => 10 => 01 => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => 01 => 10 => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => 10 => 01 => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => 001 => 100 => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => 001 => 100 => 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => 001 => 100 => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => 011 => 110 => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => 011 => 110 => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => 011 => 110 => 0 = 1 - 1
[1,5,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,5,2,4,3] => [1,2,4,3] => 001 => 100 => 0 = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[2,1] => [1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,4,1,6,3,8,5,9,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[9,7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ? = 2 - 1
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 2 - 1
[7,1,2,3,4,5,10,9,6,8] => [7,1,2,3,4,5,9,6,8] => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ?
=> ? = 3 - 1
[8,9,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[9,8,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[9,8,7,4,5,6,1,2,3] => [8,7,4,5,6,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[8,9,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[9,8,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[9,8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 3 - 1
[2,4,1,6,3,5,8,7,9] => [2,4,1,6,3,5,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[2,4,1,6,3,8,5,7,9] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,4,1,6,3,7,5,9,8] => [2,4,1,6,3,7,5,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,4,1,5,3,7,6,9,8] => [2,4,1,5,3,7,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,3,1,5,4,7,6,9,8] => [2,3,1,5,4,7,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,4,6,1,3,5,8,7,9] => [2,4,6,1,3,5,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[2,4,1,6,8,3,5,7,9] => [2,4,1,6,8,3,5,7] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[2,4,1,6,3,8,9,5,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,4,1,6,3,7,5,8,9] => [2,4,1,6,3,7,5,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,4,1,5,3,8,6,9,7] => [2,4,1,5,3,8,6,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,3,1,6,4,7,5,9,8] => [2,3,1,6,4,7,5,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[5,2,6,7,1,3,4,8,9] => [5,2,6,7,1,3,4,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,2,7,8,9,1,3,4,5] => [6,2,7,8,1,3,4,5] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[6,3,2,7,8,1,4,5,9] => [6,3,2,7,8,1,4,5] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[6,4,7,2,5,1,3,8,9] => [6,4,7,2,5,1,3,8] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[7,5,4,3,2,8,1,6,9] => [7,5,4,3,2,8,1,6] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 3 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000068
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> ([],1)
=> 1
[2,1] => [1] => [.,.]
=> ([],1)
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[6,1,2,3,4,9,8,5,7] => [6,1,2,3,4,8,5,7] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ? = 3
[6,1,2,3,4,7,8,9,5] => [6,1,2,3,4,7,8,5] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[5,1,2,3,9,8,4,6,7] => [5,1,2,3,8,4,6,7] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
=> ? = 3
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,4,1,6,3,8,5,9,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
=> ? = 2
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[7,1,2,3,4,5,10,9,6,8] => [7,1,2,3,4,5,9,6,8] => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
=> ? = 3
[10,1,2,3,4,5,11,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[10,1,2,11,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[11,10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 2
[8,9,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[9,8,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[9,8,7,4,5,6,1,2,3] => [8,7,4,5,6,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,5),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 3
[8,9,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,8,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[9,8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ? = 3
[9,8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,10,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,1,2] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,1,2,3,4,5,6,7,8,10] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[2,4,1,3,6,5,8,7,9] => [2,4,1,3,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,6),(5,7),(6,5)],8)
=> ? = 3
Description
The number of minimal elements in a poset.
Matching statistic: St000071
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> ([],1)
=> 1
[2,1] => [1] => [.,.]
=> ([],1)
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[6,1,2,3,4,9,8,5,7] => [6,1,2,3,4,8,5,7] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ? = 3
[6,1,2,3,4,7,8,9,5] => [6,1,2,3,4,7,8,5] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[5,1,2,3,9,8,4,6,7] => [5,1,2,3,8,4,6,7] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
=> ? = 3
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,4,1,6,3,8,5,9,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
=> ? = 2
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[7,1,2,3,4,5,10,9,6,8] => [7,1,2,3,4,5,9,6,8] => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
=> ? = 3
[10,1,2,3,4,5,11,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[10,1,2,11,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[11,10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 2
[8,9,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[9,8,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[9,8,7,4,5,6,1,2,3] => [8,7,4,5,6,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,5),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 3
[8,9,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,8,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[9,8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ? = 3
[9,8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,10,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,1,2] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,1,2,3,4,5,6,7,8,10] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[2,4,1,3,6,5,8,7,9] => [2,4,1,3,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,6),(5,7),(6,5)],8)
=> ? = 3
Description
The number of maximal chains in a poset.
Matching statistic: St000527
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> ([],1)
=> 1
[2,1] => [1] => [.,.]
=> ([],1)
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[6,1,2,3,4,9,8,5,7] => [6,1,2,3,4,8,5,7] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ? = 3
[6,1,2,3,4,7,8,9,5] => [6,1,2,3,4,7,8,5] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[5,1,2,3,9,8,4,6,7] => [5,1,2,3,8,4,6,7] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
=> ? = 3
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,4,1,6,3,8,5,9,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
=> ? = 2
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[7,1,2,3,4,5,10,9,6,8] => [7,1,2,3,4,5,9,6,8] => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
=> ? = 3
[10,1,2,3,4,5,11,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[10,1,2,11,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[11,10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[1,9,8,7,5,6,4,3,2] => [1,8,7,5,6,4,3,2] => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[9,8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ? = 2
[8,9,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[9,8,5,6,7,1,2,3,4] => [8,5,6,7,1,2,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3
[9,8,7,4,5,6,1,2,3] => [8,7,4,5,6,1,2,3] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,5),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 3
[8,9,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,8,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4
[9,7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[9,8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ? = 3
[9,8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,10,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,1,2] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2
[9,1,2,3,4,5,6,7,8,10] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2
[2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000196
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> 0 = 1 - 1
[2,1] => [1] => [.,.]
=> 0 = 1 - 1
[1,2,3] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ? = 2 - 1
[9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ? = 2 - 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 2 - 1
[6,1,2,3,4,9,8,5,7] => [6,1,2,3,4,8,5,7] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 3 - 1
[6,1,2,3,4,7,8,9,5] => [6,1,2,3,4,7,8,5] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 2 - 1
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[5,1,2,3,9,8,4,6,7] => [5,1,2,3,8,4,6,7] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 3 - 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[2,4,1,6,3,8,5,9,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ? = 4 - 1
[9,3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[9,7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 3 - 1
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 1 - 1
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 1 - 1
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? = 1 - 1
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? = 1 - 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ? = 2 - 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 2 - 1
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 2 - 1
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 2 - 1
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 2 - 1
[7,1,2,3,4,5,10,9,6,8] => [7,1,2,3,4,5,9,6,8] => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? = 3 - 1
[10,1,2,3,4,5,11,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 2 - 1
[10,1,2,11,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 2 - 1
[11,10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 2 - 1
[1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 1 - 1
[1,9,8,7,5,6,4,3,2] => [1,8,7,5,6,4,3,2] => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ? = 2 - 1
[1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ? = 1 - 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ? = 1 - 1
[9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 2 - 1
[10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 2 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000632
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> ([],1)
=> 0 = 1 - 1
[2,1] => [1] => [.,.]
=> ([],1)
=> 0 = 1 - 1
[1,2,3] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,3,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,3,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 0 = 1 - 1
[3,1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[9,7,6,5,4,3,2,10,1,8] => [9,7,6,5,4,3,2,1,8] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,5,8,6] => [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[6,1,2,3,4,9,8,5,7] => [6,1,2,3,4,8,5,7] => [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ? = 3 - 1
[6,1,2,3,4,7,8,9,5] => [6,1,2,3,4,7,8,5] => [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[8,1,2,3,9,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[5,1,2,3,9,8,4,6,7] => [5,1,2,3,8,4,6,7] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
=> ? = 3 - 1
[2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[2,4,1,6,3,8,5,9,7] => [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 4 - 1
[9,3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[9,7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,7),(1,3),(2,4),(3,7),(4,5),(5,6),(7,6)],8)
=> ? = 3 - 1
[2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1 - 1
[2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1 - 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ([(0,10),(1,9),(2,10),(3,4),(4,6),(5,3),(6,8),(7,5),(8,2),(9,7)],11)
=> ? = 2 - 1
[2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2 - 1
[9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,9,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,5,9,4,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[8,7,6,5,4,3,1,2,9] => [8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[9,1,10,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[10,1,2,3,4,5,6,7,8,11,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2 - 1
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
[7,1,2,3,4,5,10,9,6,8] => [7,1,2,3,4,5,9,6,8] => [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,4),(6,3),(8,5)],9)
=> ? = 3 - 1
[10,1,2,3,4,5,11,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2 - 1
[10,1,2,11,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2 - 1
[11,10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,9),(1,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],10)
=> ? = 2 - 1
[1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1 - 1
[1,9,8,7,5,6,4,3,2] => [1,8,7,5,6,4,3,2] => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2 - 1
[1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1 - 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 1 - 1
[9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 2 - 1
[10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
=> ? = 2 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000353
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 75%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1] => [1] => ? = 1 - 1
[2,1] => [1] => [1] => ? = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [2,1] => 0 = 1 - 1
[2,3,1] => [2,1] => [2,1] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,2,4,3] => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,2,4,3] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,2,4,3] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 - 1
[7,8,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 2 - 1
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[7,5,6,8,4,3,2,1] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 2 - 1
[6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 2 - 1
[8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 2 - 1
[7,8,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 2 - 1
[8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 2 - 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 2 - 1
[7,6,4,5,8,3,2,1] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 2 - 1
[7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => ? = 2 - 1
[7,4,5,6,8,3,2,1] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 2 - 1
[8,7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 2 - 1
[7,8,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 2 - 1
[8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2 - 1
[8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 3 - 1
[6,7,8,3,4,5,2,1] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 2 - 1
[8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [7,3,2,1,6,5,4] => ? = 2 - 1
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[6,4,5,7,3,8,2,1] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => ? = 2 - 1
[5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 2 - 1
[7,6,5,3,4,8,2,1] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 2 - 1
[7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => ? = 2 - 1
[7,6,3,4,5,8,2,1] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 2 - 1
[6,3,4,5,7,8,2,1] => [6,3,4,5,7,2,1] => [4,1,2,3,7,6,5] => ? = 2 - 1
[4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ? = 2 - 1
[8,7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => ? = 2 - 1
[7,8,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => ? = 2 - 1
[8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 2 - 1
[8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => [7,2,6,5,1,4,3] => ? = 3 - 1
[8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => [7,6,2,5,1,4,3] => ? = 3 - 1
[8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => [3,7,2,6,1,5,4] => ? = 3 - 1
[8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [7,6,5,2,1,4,3] => ? = 2 - 1
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [4,3,7,2,1,6,5] => ? = 3 - 1
[5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [1,4,7,3,2,6,5] => ? = 2 - 1
[8,7,6,5,2,3,4,1] => [7,6,5,2,3,4,1] => [7,6,5,1,2,4,3] => ? = 2 - 1
[6,7,8,5,2,3,4,1] => [6,7,5,2,3,4,1] => [3,7,6,1,2,5,4] => ? = 2 - 1
[8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [2,4,7,1,3,6,5] => ? = 2 - 1
[5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => [2,4,7,1,3,6,5] => ? = 2 - 1
[7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [7,2,1,3,4,6,5] => ? = 2 - 1
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1 - 1
[6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 - 1
[6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 2 - 1
[6,4,5,7,3,2,8,1] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => ? = 2 - 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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