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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St000353
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000834
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,2,3] => 1 = 0 + 1
[2,1,3] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,3,2] => 1 = 0 + 1
[3,2,1] => [1,3,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => 1 = 0 + 1
[1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => 1 = 0 + 1
[3,4,1,2] => [1,3,2,4] => 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => 1 = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => 1 = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => 2 = 1 + 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000196
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 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: St000291
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => 01 => 0
[3,2,1] => [1,3,2] => 01 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => 001 => 0
[1,4,3,2] => [1,2,4,3] => 001 => 0
[2,1,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => 001 => 0
[2,4,3,1] => [1,2,4,3] => 001 => 0
[3,1,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => 001 => 0
[3,2,1,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => 001 => 0
[3,4,1,2] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => 011 => 0
[4,1,3,2] => [1,4,2,3] => 010 => 1
[4,2,1,3] => [1,4,3,2] => 011 => 0
[4,2,3,1] => [1,4,2,3] => 010 => 1
[4,3,1,2] => [1,4,2,3] => 010 => 1
[4,3,2,1] => [1,4,2,3] => 010 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 0
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 0
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 0
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1
Description
The number of descents of a binary word.
Matching statistic: St000201
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 1 = 0 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 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: St000023
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1] => 0
[2,1] => [1,2] => [2,1] => [1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [2,1] => 0
[1,3,2] => [1,2,3] => [3,2,1] => [2,1] => 0
[2,1,3] => [1,2,3] => [3,2,1] => [2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [2,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [1,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [2,1,3] => 0
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [2,1,3] => 0
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [1,2,3] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [4,2,1,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [4,2,1,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2] => 0 => 0 => 0
[2,1] => [1,2] => 0 => 0 => 0
[1,2,3] => [1,2,3] => 00 => 00 => 0
[1,3,2] => [1,2,3] => 00 => 00 => 0
[2,1,3] => [1,2,3] => 00 => 00 => 0
[2,3,1] => [1,2,3] => 00 => 00 => 0
[3,1,2] => [1,3,2] => 01 => 10 => 0
[3,2,1] => [1,3,2] => 01 => 10 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0
[1,2,4,3] => [1,2,3,4] => 000 => 000 => 0
[1,3,2,4] => [1,2,3,4] => 000 => 000 => 0
[1,3,4,2] => [1,2,3,4] => 000 => 000 => 0
[1,4,2,3] => [1,2,4,3] => 001 => 100 => 0
[1,4,3,2] => [1,2,4,3] => 001 => 100 => 0
[2,1,3,4] => [1,2,3,4] => 000 => 000 => 0
[2,1,4,3] => [1,2,3,4] => 000 => 000 => 0
[2,3,1,4] => [1,2,3,4] => 000 => 000 => 0
[2,3,4,1] => [1,2,3,4] => 000 => 000 => 0
[2,4,1,3] => [1,2,4,3] => 001 => 100 => 0
[2,4,3,1] => [1,2,4,3] => 001 => 100 => 0
[3,1,2,4] => [1,3,2,4] => 010 => 010 => 1
[3,1,4,2] => [1,3,4,2] => 001 => 100 => 0
[3,2,1,4] => [1,3,2,4] => 010 => 010 => 1
[3,2,4,1] => [1,3,4,2] => 001 => 100 => 0
[3,4,1,2] => [1,3,2,4] => 010 => 010 => 1
[3,4,2,1] => [1,3,2,4] => 010 => 010 => 1
[4,1,2,3] => [1,4,3,2] => 011 => 110 => 0
[4,1,3,2] => [1,4,2,3] => 010 => 010 => 1
[4,2,1,3] => [1,4,3,2] => 011 => 110 => 0
[4,2,3,1] => [1,4,2,3] => 010 => 010 => 1
[4,3,1,2] => [1,4,2,3] => 010 => 010 => 1
[4,3,2,1] => [1,4,2,3] => 010 => 010 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1000 => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1000 => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1000 => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1000 => 0
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 0100 => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1000 => 0
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 0100 => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1000 => 0
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 0100 => 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 0100 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000632
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 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: St001037
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,1,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[3,2,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,2,1,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,4,1,2] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[4,2,3,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001307The number of induced stars on four vertices in a graph. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000099The number of valleys of a permutation, including the boundary. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000659The number of rises of length at least 2 of a Dyck path. St001280The number of parts of an integer partition that are at least two. St000312The number of leaves in a graph. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000035The number of left outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000092The number of outer peaks of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000523The number of 2-protected nodes of a rooted tree. St001964The interval resolution global dimension of a poset. St001960The number of descents of a permutation minus one if its first entry is not one. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001868The number of alignments of type NE of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001946The number of descents in a parking function. St000679The pruning number of an ordered tree.
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