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Your data matches 237 different statistics following compositions of up to 3 maps.
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Matching statistic: St000836
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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] => 1
[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] => 1
[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] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[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]
=> [3,4,2,1] => 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] => 1
[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] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[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]
=> [1,4,5,3,2] => 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] => 1
[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] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[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]
=> [2,4,5,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000837
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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] => 1
[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] => 1
[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] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[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] => 1
[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] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1
[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] => 1
[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] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1
[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
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St001405
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3 = 2 + 1
Description
The number of bonds in a permutation.
For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St000021
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000204
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 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: St000354
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000829
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St001083
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
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: St001489
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St001683
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,5,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
The following 227 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000225Difference between largest and smallest parts in a partition. St000741The Colin de Verdière graph invariant. St001315The dissociation number of a graph. St001933The largest multiplicity of a part in an integer partition. St000392The length of the longest run of ones in a binary word. St000306The bounce count of a Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000015The number of peaks of a Dyck path. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000624The normalized sum of the minimal distances to a greater element. St000993The multiplicity of the largest part of an integer partition. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001469The holeyness of a permutation. St000668The least common multiple of the parts of the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000353The number of inner valleys of a permutation. St000710The number of big deficiencies of a permutation. St001114The number of odd descents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000871The number of very big ascents of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001924The number of cells in an integer partition whose arm and leg length coincide. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001568The smallest positive integer that does not appear twice in the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000308The height of the tree associated to a permutation. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000456The monochromatic index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001388The number of non-attacking neighbors of a permutation. St000402Half the size of the symmetry class of a permutation. St000628The balance of a binary word. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000877The depth of the binary word interpreted as a path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001118The acyclic chromatic index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001982The number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001520The number of strict 3-descents. St001330The hat guessing number of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001820The size of the image of the pop stack sorting operator. St000292The number of ascents of a binary word. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000742The number of big ascents of a permutation after prepending zero. St001638The book thickness of a graph. St001487The number of inner corners of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000646The number of big ascents of a permutation. St000035The number of left outer peaks of a permutation. St001052The length of the exterior of a permutation. St001060The distinguishing index of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001875The number of simple modules with projective dimension at most 1. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001877Number of indecomposable injective modules with projective dimension 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000872The number of very big descents of a permutation. St000929The constant term of the character polynomial of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001964The interval resolution global dimension of a poset. St001644The dimension of a graph. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000761The number of ascents in an integer composition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000647The number of big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001423The number of distinct cubes in a binary word. St001556The number of inversions of the third entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001715The number of non-records in a permutation. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001729The number of visible descents of a permutation. St000381The largest part of an integer composition. St000904The maximal number of repetitions of an integer composition. St001415The length of the longest palindromic prefix of a binary word. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition.
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