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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000371
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001229
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001229: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001229: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
The vector space dimension of the first extension group between the Jacobson radical J and J^2.
The vector space dimension of $Ext_A^1(J,J^2)$.
Matching statistic: St001682
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001682: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001682: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 2
Description
The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.
Matching statistic: St000316
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 2
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000372
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [5,2,3,1,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [5,1,3,2,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,1,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,5,1,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1,3,2,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1,2,3,5] => 2
Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Matching statistic: St000024
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000204
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 2
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000356
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [5,1,3,2,4] => [5,1,4,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [4,5,2,1,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,5,1,2,3] => [4,5,1,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [4,3,5,1,2] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [4,3,2,5,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,3,1,2,5] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,5,1] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,1,5] => [4,2,5,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1,3,2,5] => [4,1,5,3,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1,2,3,5] => [4,1,5,3,2] => 2
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St001083
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,2,3] => [4,1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => [3,2,4,1] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => [3,1,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,4,1] => [2,4,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [5,1,3,2,4] => [5,1,4,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,1,2,3,4] => [5,1,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [4,5,2,1,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,5,1,2,3] => [4,5,1,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [4,3,5,1,2] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [4,3,2,5,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,3,1,2,5] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,5,1] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,1,5] => [4,2,5,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1,3,2,5] => [4,1,5,3,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1,2,3,5] => [4,1,5,3,2] => 2
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001089
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001089: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000443The number of long tunnels of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001082The number of boxed occurrences of 123 in a permutation. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001725The harmonious chromatic number of a graph. St000456The monochromatic index of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001480The number of simple summands of the module J^2/J^3. St000005The bounce statistic of a Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St001330The hat guessing number of a graph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001866The nesting alignments of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001862The number of crossings of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001905The number of preferred parking spots in a parking function less than the index of the car. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
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