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Your data matches 121 different statistics following compositions of up to 3 maps.
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Matching statistic: St000836
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00025: Dyck paths —to 132-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]
=> [2,1] => 0
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[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] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[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] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
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 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00031: Dyck paths —to 312-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]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[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] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[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] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
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: St001388
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001388: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001388: 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] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => 0
[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] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[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] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[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] => 2
[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] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[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] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[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] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[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] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[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] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[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] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3
[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 non-attacking neighbors of a permutation.
For a permutation $\sigma$, the indices $i$ and $i+1$ are attacking if $|\sigma(i)-\sigma(i+1)| = 1$.
Visually, this is, for $\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns $i$ and $i+1$ are non-attacking.
Matching statistic: St001315
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => ([],2)
=> 2 = 0 + 2
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
Description
The dissociation number of a graph.
Matching statistic: St001087
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001087: 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
St001087: 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] => 0
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[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] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 2
[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] => 2
[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] => 2
[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] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2
[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] => 2
[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] => 2
[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] => 2
[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] => 1
[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] => 2
[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] => 2
[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] => 3
[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] => 1
[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] => 2
[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] => 1
[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] => 3
[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] => 2
[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] => 0
[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] => 3
[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] => 2
[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] => 2
[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] => 3
[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] => 1
[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] => 3
[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] => 2
[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] => 2
[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] => 2
[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] => 1
[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] => 2
[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] => 2
[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] => 1
[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] => 1
[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] => 2
[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] => 2
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St000245
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
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%
Mp00239: Permutations —Corteel⟶ Permutations
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%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 2 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 3 = 2 + 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 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 4 = 3 + 1
[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 + 1
[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] => 3 = 2 + 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 + 1
[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] => 4 = 3 + 1
[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] => 3 = 2 + 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] => 1 = 0 + 1
[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] => 4 = 3 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 4 = 3 + 1
[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
[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] => 4 = 3 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 3 = 2 + 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
[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] => 3 = 2 + 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] => 3 = 2 + 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
[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
[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] => 3 = 2 + 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] => 3 = 2 + 1
Description
The number of ascents of a permutation.
Matching statistic: St000291
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1010 => 1101 => 0101 => 1 = 0 + 1
[1,1,0,0]
=> 1100 => 1101 => 0101 => 1 = 0 + 1
[1,0,1,0,1,0]
=> 101010 => 110101 => 010101 => 2 = 1 + 1
[1,0,1,1,0,0]
=> 101100 => 110101 => 010101 => 2 = 1 + 1
[1,1,0,0,1,0]
=> 110010 => 110101 => 010101 => 2 = 1 + 1
[1,1,0,1,0,0]
=> 110100 => 111001 => 011001 => 1 = 0 + 1
[1,1,1,0,0,0]
=> 111000 => 111001 => 011001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11010101 => 01010101 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11010101 => 01010101 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11010101 => 01010101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11011001 => 01011001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11011001 => 01011001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 11010101 => 01010101 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 11010101 => 01010101 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 11100101 => 01100101 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => 01101001 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 11101001 => 01101001 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11100101 => 01100101 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11101001 => 01101001 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11110001 => 01110001 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11110001 => 01110001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 1101011001 => 0101011001 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 1101011001 => 0101011001 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1101100101 => 0101100101 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 1101101001 => 0101101001 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 1101101001 => 0101101001 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1101100101 => 0101100101 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 1101101001 => 0101101001 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 1101110001 => 0101110001 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1101110001 => 0101110001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1101010101 => 0101010101 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1101011001 => 0101011001 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1101011001 => 0101011001 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1110010101 => 0110010101 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1110010101 => 0110010101 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1110100101 => 0110100101 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1110101001 => 0110101001 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1110101001 => 0110101001 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1110100101 => 0110100101 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1110101001 => 0110101001 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1110110001 => 0110110001 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1110110001 => 0110110001 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1110010101 => 0110010101 => 3 = 2 + 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1010 => 0101 => 1010 => 1 = 0 + 1
[1,1,0,0]
=> 1100 => 1001 => 1010 => 1 = 0 + 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 101010 => 2 = 1 + 1
[1,0,1,1,0,0]
=> 101100 => 011001 => 101010 => 2 = 1 + 1
[1,1,0,0,1,0]
=> 110010 => 100101 => 101010 => 2 = 1 + 1
[1,1,0,1,0,0]
=> 110100 => 101001 => 110010 => 1 = 0 + 1
[1,1,1,0,0,0]
=> 111000 => 110001 => 110010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 10101010 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01011001 => 10101010 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 01100101 => 10101010 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 01101001 => 10110010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 01110001 => 10110010 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10010101 => 10101010 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 10101010 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10100101 => 11001010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10101001 => 11010010 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10110001 => 11010010 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11000101 => 11001010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11001001 => 11010010 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11010001 => 11100010 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11100001 => 11100010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 1010101010 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101011001 => 1010101010 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101100101 => 1010101010 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101101001 => 1010110010 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101110001 => 1010110010 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0110010101 => 1010101010 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0110011001 => 1010101010 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0110100101 => 1011001010 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0110101001 => 1011010010 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0110110001 => 1011010010 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0111000101 => 1011001010 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0111001001 => 1011010010 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0111010001 => 1011100010 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0111100001 => 1011100010 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001010101 => 1010101010 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1001011001 => 1010101010 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1001100101 => 1010101010 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1001101001 => 1010110010 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1001110001 => 1010110010 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1010010101 => 1100101010 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1010011001 => 1100101010 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1010100101 => 1101001010 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1010101001 => 1101010010 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1010110001 => 1101010010 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1011000101 => 1101001010 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1011001001 => 1101010010 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1011010001 => 1101100010 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1011100001 => 1101100010 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1100010101 => 1100101010 => 3 = 2 + 1
Description
The number of ascents of a binary word.
Matching statistic: St000619
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => [2,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [4,1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => [3,1,2,5,4] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [5,3,1,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [2,1,4,3,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,4,2] => [5,2,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,1,5,2] => [3,2,1,5,4] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,1,3,2] => [2,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [5,4,3,1,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [5,2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [3,2,1,5,4] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,4,2,1,3] => [5,4,2,1,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [5,1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [5,1,2,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [3,1,2,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,5,3,1] => [2,1,5,4,3] => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [5,4,1,3,2] => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 3 = 2 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000672: 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
St000672: 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] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,2,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,2,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 2 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 3 = 2 + 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 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 4 = 3 + 1
[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 + 1
[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] => 3 = 2 + 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 + 1
[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] => 4 = 3 + 1
[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] => 3 = 2 + 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] => 1 = 0 + 1
[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] => 4 = 3 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 4 = 3 + 1
[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
[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] => 4 = 3 + 1
[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] => 3 = 2 + 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] => 3 = 2 + 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] => 3 = 2 + 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
[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] => 3 = 2 + 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] => 3 = 2 + 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
[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
[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] => 3 = 2 + 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] => 3 = 2 + 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
The following 111 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000062The length of the longest increasing subsequence of the permutation. St000203The number of external nodes of a binary tree. St000308The height of the tree associated to a permutation. St000390The number of runs of ones in a binary word. St000991The number of right-to-left minima of a permutation. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for 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). St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001298The number of repeated entries in the Lehmer code of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000454The largest eigenvalue of a graph if it is integral. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000331The number of upper interactions of a Dyck path. St001394The genus of a permutation. St000144The pyramid weight of the Dyck path. 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 $eA$. 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 $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000474Dyson's crank of a partition. St001432The order dimension of the partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000477The weight of a partition according to Alladi. 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. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. 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. St000870The product of the hook lengths of the diagonal cells in 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. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001060The distinguishing index of a graph. St000670The reversal length of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000893The number of distinct diagonal sums of an alternating sign matrix. St000353The number of inner valleys of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000013The height of a Dyck path. St001488The number of corners of a skew partition. St001618The cardinality of the Frattini sublattice of a 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000742The number of big ascents of a permutation after prepending zero. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000632The jump number of the poset. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000028The number of stack-sorts needed to sort a permutation. St000871The number of very big ascents of a permutation. St000035The number of left outer peaks of a permutation. St001638The book thickness of a graph. St000264The girth of a graph, which is not a tree. St000538The number of even inversions of a permutation. St000808The number of up steps of the associated bargraph. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001822The number of alignments of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000092The number of outer peaks of a permutation. St000306The bounce count of a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001557The number of inversions of the second entry of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001637The number of (upper) dissectors of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St001330The hat guessing number of a graph. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000628The balance of a binary word. St000640The rank of the largest boolean interval in a poset. St000386The number of factors DDU in a Dyck path.
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