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Your data matches 141 different statistics following compositions of up to 3 maps.
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Matching statistic: St000023
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => 0
[1,2] => [1] => 0
[2,1] => [1] => 0
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 0
[2,1,4,3] => [2,1,3] => 0
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 0
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 0
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[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] => 0
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[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] => 1
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[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] => 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000035
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [1,2] => 0
[2,3,1] => [2,1] => [1,2] => 0
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [1,2] => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,2,3] => 0
[1,3,4,2] => [1,3,2] => [1,2,3] => 0
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3,4] => [2,1,3] => [1,2,3] => 0
[2,1,4,3] => [2,1,3] => [1,2,3] => 0
[2,3,1,4] => [2,3,1] => [1,2,3] => 0
[2,3,4,1] => [2,3,1] => [1,2,3] => 0
[2,4,1,3] => [2,1,3] => [1,2,3] => 0
[2,4,3,1] => [2,3,1] => [1,2,3] => 0
[3,1,2,4] => [3,1,2] => [1,3,2] => 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 1
[3,2,1,4] => [3,2,1] => [1,3,2] => 1
[3,2,4,1] => [3,2,1] => [1,3,2] => 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 1
[3,4,2,1] => [3,2,1] => [1,3,2] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,2,3] => 0
[4,2,1,3] => [2,1,3] => [1,2,3] => 0
[4,2,3,1] => [2,3,1] => [1,2,3] => 0
[4,3,1,2] => [3,1,2] => [1,3,2] => 1
[4,3,2,1] => [3,2,1] => [1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => 0
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => 0
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => 1
Description
The number of left outer peaks of a permutation.
A left 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 $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000196
(load all 4 compositions to match this statistic)
(load all 4 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] => [1] => [.,.]
=> 0
[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
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: St000994
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [1,2] => 0
[2,3,1] => [2,1] => [1,2] => 0
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [1,2] => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,2,3] => 0
[1,3,4,2] => [1,3,2] => [1,2,3] => 0
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3,4] => [2,1,3] => [1,2,3] => 0
[2,1,4,3] => [2,1,3] => [1,2,3] => 0
[2,3,1,4] => [2,3,1] => [1,2,3] => 0
[2,3,4,1] => [2,3,1] => [1,2,3] => 0
[2,4,1,3] => [2,1,3] => [1,2,3] => 0
[2,4,3,1] => [2,3,1] => [1,2,3] => 0
[3,1,2,4] => [3,1,2] => [1,3,2] => 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 1
[3,2,1,4] => [3,2,1] => [1,3,2] => 1
[3,2,4,1] => [3,2,1] => [1,3,2] => 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 1
[3,4,2,1] => [3,2,1] => [1,3,2] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,2,3] => 0
[4,2,1,3] => [2,1,3] => [1,2,3] => 0
[4,2,3,1] => [2,3,1] => [1,2,3] => 0
[4,3,1,2] => [3,1,2] => [1,3,2] => 1
[4,3,2,1] => [3,2,1] => [1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => 0
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => 0
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000201
(load all 4 compositions to match this statistic)
(load all 4 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] => [1] => [.,.]
=> 1 = 0 + 1
[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
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: St000386
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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] => [1] => [.,.]
=> [1,0]
=> 0
[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
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000632
(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
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] => [1] => [.,.]
=> ([],1)
=> 0
[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
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: St000834
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => [] => 0
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [2,1] => 0
[1,3,2] => [1,2] => [1,2] => [2,1] => 0
[2,1,3] => [2,1] => [1,2] => [2,1] => 0
[2,3,1] => [2,1] => [1,2] => [2,1] => 0
[3,1,2] => [1,2] => [1,2] => [2,1] => 0
[3,2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2,4] => [1,3,2] => [1,2,3] => [3,2,1] => 0
[1,3,4,2] => [1,3,2] => [1,2,3] => [3,2,1] => 0
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,4,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 0
[2,1,3,4] => [2,1,3] => [1,2,3] => [3,2,1] => 0
[2,1,4,3] => [2,1,3] => [1,2,3] => [3,2,1] => 0
[2,3,1,4] => [2,3,1] => [1,2,3] => [3,2,1] => 0
[2,3,4,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0
[2,4,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 0
[2,4,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2,4] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[3,1,4,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[3,2,1,4] => [3,2,1] => [1,3,2] => [2,3,1] => 1
[3,2,4,1] => [3,2,1] => [1,3,2] => [2,3,1] => 1
[3,4,1,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[3,4,2,1] => [3,2,1] => [1,3,2] => [2,3,1] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[4,1,3,2] => [1,3,2] => [1,2,3] => [3,2,1] => 0
[4,2,1,3] => [2,1,3] => [1,2,3] => [3,2,1] => 0
[4,2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 0
[4,3,1,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[4,3,2,1] => [3,2,1] => [1,3,2] => [2,3,1] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 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: 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] => [1] => [1] => [1,0]
=> 0
[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
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001307
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001307: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001307: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 0
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
Description
The number of induced stars on four vertices in a graph.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000021The number of descents of a permutation. St000291The number of descents of a binary word. St000884The number of isolated descents of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000523The number of 2-protected nodes of a rooted tree. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000353The number of inner valleys of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St001128The exponens consonantiae of a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001638The book thickness of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001323The independence gap of a graph. St000260The radius of a connected graph. St000252The number of nodes of degree 3 of a binary tree. St001470The cyclic holeyness of a permutation. St001964The interval resolution global dimension of a poset. St001487The number of inner corners of a skew partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000862The number of parts of the shifted shape of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000259The diameter 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001845The number of join irreducibles minus the rank of a lattice. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001866The nesting alignments 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001490The number of connected components of a skew partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001694The number of maximal dissociation sets in a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra 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 $K[x]/(x^n)$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001768The number of reduced words of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001857The number of edges in the reduced word graph of a signed permutation. St000322The skewness of a graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.
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