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Your data matches 166 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,2,3] => 2
[2,1,3] => [1,2,3] => 2
[2,3,1] => [1,2,3] => 2
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 3
[1,2,4,3] => [1,2,3,4] => 3
[1,3,2,4] => [1,2,3,4] => 3
[1,3,4,2] => [1,2,3,4] => 3
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 3
[2,1,4,3] => [1,2,3,4] => 3
[2,3,1,4] => [1,2,3,4] => 3
[2,3,4,1] => [1,2,3,4] => 3
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 4
[1,2,3,5,4] => [1,2,3,4,5] => 4
[1,2,4,3,5] => [1,2,3,4,5] => 4
[1,2,4,5,3] => [1,2,3,4,5] => 4
[1,2,5,3,4] => [1,2,3,5,4] => 3
[1,2,5,4,3] => [1,2,3,5,4] => 3
[1,3,2,4,5] => [1,2,3,4,5] => 4
[1,3,2,5,4] => [1,2,3,4,5] => 4
[1,3,4,2,5] => [1,2,3,4,5] => 4
[1,3,4,5,2] => [1,2,3,4,5] => 4
[1,3,5,2,4] => [1,2,3,5,4] => 3
[1,3,5,4,2] => [1,2,3,5,4] => 3
[1,4,2,3,5] => [1,2,4,3,5] => 3
[1,4,2,5,3] => [1,2,4,5,3] => 3
[1,4,3,2,5] => [1,2,4,3,5] => 3
[1,4,3,5,2] => [1,2,4,5,3] => 3
[1,4,5,2,3] => [1,2,4,3,5] => 3
Description
The number of ascents of a permutation.
Matching statistic: St001298
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,2,3] => 2
[2,1,3] => [1,2,3] => 2
[2,3,1] => [1,2,3] => 2
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 3
[1,2,4,3] => [1,2,3,4] => 3
[1,3,2,4] => [1,2,3,4] => 3
[1,3,4,2] => [1,2,3,4] => 3
[1,4,2,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => 3
[2,1,4,3] => [1,2,3,4] => 3
[2,3,1,4] => [1,2,3,4] => 3
[2,3,4,1] => [1,2,3,4] => 3
[2,4,1,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 4
[1,2,3,5,4] => [1,2,3,4,5] => 4
[1,2,4,3,5] => [1,2,3,4,5] => 4
[1,2,4,5,3] => [1,2,3,4,5] => 4
[1,2,5,3,4] => [1,2,3,5,4] => 3
[1,2,5,4,3] => [1,2,3,5,4] => 3
[1,3,2,4,5] => [1,2,3,4,5] => 4
[1,3,2,5,4] => [1,2,3,4,5] => 4
[1,3,4,2,5] => [1,2,3,4,5] => 4
[1,3,4,5,2] => [1,2,3,4,5] => 4
[1,3,5,2,4] => [1,2,3,5,4] => 3
[1,3,5,4,2] => [1,2,3,5,4] => 3
[1,4,2,3,5] => [1,2,4,3,5] => 3
[1,4,2,5,3] => [1,2,4,5,3] => 3
[1,4,3,2,5] => [1,2,4,3,5] => 3
[1,4,3,5,2] => [1,2,4,5,3] => 3
[1,4,5,2,3] => [1,2,4,3,5] => 3
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St000991
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 2 = 1 + 1
[2,1] => [1,2] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 3 = 2 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => 3 = 2 + 1
[3,1,2] => [1,2,3] => 3 = 2 + 1
[3,2,1] => [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[1,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [1,4,2,3] => 3 = 2 + 1
[2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => 3 = 2 + 1
[2,3,1,4] => [1,4,2,3] => 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[2,4,1,3] => [1,3,2,4] => 3 = 2 + 1
[2,4,3,1] => [1,2,4,3] => 3 = 2 + 1
[3,1,2,4] => [1,2,4,3] => 3 = 2 + 1
[3,1,4,2] => [1,4,2,3] => 3 = 2 + 1
[3,2,1,4] => [1,4,2,3] => 3 = 2 + 1
[3,2,4,1] => [1,2,4,3] => 3 = 2 + 1
[3,4,1,2] => [1,2,3,4] => 4 = 3 + 1
[3,4,2,1] => [1,2,3,4] => 4 = 3 + 1
[4,1,2,3] => [1,2,3,4] => 4 = 3 + 1
[4,1,3,2] => [1,3,2,4] => 3 = 2 + 1
[4,2,1,3] => [1,3,2,4] => 3 = 2 + 1
[4,2,3,1] => [1,2,3,4] => 4 = 3 + 1
[4,3,1,2] => [1,2,3,4] => 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,2,4,5,3] => [1,2,4,5,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,3,4] => 4 = 3 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => 4 = 3 + 1
[1,4,2,5,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => 4 = 3 + 1
[1,4,5,2,3] => [1,4,5,2,3] => 3 = 2 + 1
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000021
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,2,3] => [3,2,1] => 2
[2,1,3] => [1,2,3] => [3,2,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 2
[3,1,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 3
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 3
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 3
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 3
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 3
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: St000024
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 2
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
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: St000155
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,2,3] => [2,3,1] => 2
[2,1,3] => [1,2,3] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,3,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => 2
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 2
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 2
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => 2
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => 2
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 2
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => 2
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 2
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => 2
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 3
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => 3
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => 3
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000394
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 2
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000662
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,2,3] => [3,2,1] => 2
[2,1,3] => [1,2,3] => [3,2,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 2
[3,1,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 3
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 3
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 3
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 3
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 3
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 2
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => 3
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000996
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,3,2] => [3,2,1] => 1
[2,1,3] => [1,3,2] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => 2
[3,1,2] => [1,2,3] => [2,3,1] => 2
[3,2,1] => [1,2,3] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 2
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => 2
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => 1
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => 2
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 3
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 2
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => 2
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => 2
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => 2
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => 2
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => 3
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => 3
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 3
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => 3
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 3
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 3
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => 3
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001189
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(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 2
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 2
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
The following 156 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000007The number of saliances of the permutation. St000010The length of the partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000272The treewidth of a graph. St000288The number of ones in a binary word. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000552The number of cut vertices of a graph. St000703The number of deficiencies of a permutation. St000784The maximum of the length and the largest part of the integer partition. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. 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. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001692The number of vertices with higher degree than the average degree in a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000308The height of the tree associated to a permutation. St000378The diagonal inversion number of an integer partition. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. 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. St001202Call 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 special CNakayama 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:
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. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001654The monophonic hull number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck 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. St000619The number of cyclic descents of a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001480The number of simple summands of the module J^2/J^3. St000691The number of changes of a binary word. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St000829The Ulam distance of a permutation to the identity permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000711The number of big exceedences of a permutation. St000068The number of minimal elements in a poset. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001427The number of descents of a signed permutation. St001812The biclique partition number of a graph. St000259The diameter of a connected graph. St001644The dimension of a graph. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001820The size of the image of the pop stack sorting operator. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001649The length of a longest trail in a graph. St001863The number of weak excedances of a signed permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001960The number of descents of a permutation minus one if its first entry is not one. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001626The number of maximal proper sublattices of a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001946The number of descents in a parking function. St000942The number of critical left to right maxima of the parking functions. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001642The Prague dimension of a graph. St000640The rank of the largest boolean interval in a poset.
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