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Your data matches 100 different statistics following compositions of up to 3 maps.
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Matching statistic: St000540
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000540: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 1
[2,1] => 1
[1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[2,1,1] => 1
[1,1,3] => 2
[1,3,1] => 2
[3,1,1] => 2
[1,2,2] => 2
[2,1,2] => 2
[2,2,1] => 2
[1,2,3] => 3
[1,3,2] => 3
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 3
[1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 1
[1,1,1,3] => 2
[1,1,3,1] => 2
[1,3,1,1] => 2
[3,1,1,1] => 2
[1,1,1,4] => 3
[1,1,4,1] => 3
[1,4,1,1] => 3
[4,1,1,1] => 3
[1,1,2,2] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[2,1,1,2] => 2
[2,1,2,1] => 2
[2,2,1,1] => 2
[1,1,2,3] => 3
[1,1,3,2] => 3
[1,2,1,3] => 3
[1,2,3,1] => 3
[1,3,1,2] => 3
[1,3,2,1] => 3
[2,1,1,3] => 3
[2,1,3,1] => 3
[2,3,1,1] => 3
[3,1,1,2] => 3
[3,1,2,1] => 3
Description
The sum of the entries of a parking function minus its length.
Under the standard map to labelled rooted forests, this statistic is sent to the number of inversions.
Matching statistic: St000018
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,1,0,0]
=> [1,2] => 0
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,0,1,0]
=> [2,1] => 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000228
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> 0
[1,1] => [1,1,0,0]
=> []
=> 0
[1,2] => [1,0,1,0]
=> [1]
=> 1
[2,1] => [1,0,1,0]
=> [1]
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> []
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [1]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [2]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [2]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000246
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,1,0,0]
=> [2,1] => 0
[1,2] => [1,0,1,0]
=> [1,2] => 1
[2,1] => [1,0,1,0]
=> [1,2] => 1
[1,1,1] => [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,1,2] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000719
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000719: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000719: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [(1,2)]
=> 0
[1,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 0
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[2,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
Description
The number of alignments in a perfect matching.
An alignment is a pair of edges $(i,j)$, $(k,l)$ such that $i < j < k < l$.
Since any two edges in a perfect matching are either nesting ([[St000041]]), crossing ([[St000042]]) or form an alignment, the sum of these numbers in a perfect matching with $n$ edges is $\binom{n}{2}$.
Matching statistic: St001759
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,1,0,0]
=> [1,2] => 0
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,0,1,0]
=> [2,1] => 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
Description
The Rajchgot index of a permutation.
The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.
Matching statistic: St000004
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 0
[1,1] => [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,2] => [1,0,1,0]
=> [2,1] => [2,1] => 1
[2,1] => [1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000057
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[1],[2]]
=> [[1,2]]
=> 0
[1,1] => [1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
[1,2] => [1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[2,1] => [1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3],[2,4],[5,6]]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3],[2,4],[5,6]]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3],[2,4],[5,6]]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,4],[2,6],[3,7],[5,8]]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,4],[2,6],[3,7],[5,8]]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,4],[2,6],[3,7],[5,8]]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,4],[2,6],[3,7],[5,8]]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,4],[2,5],[3,7],[6,8]]
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,4],[2,5],[3,7],[6,8]]
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,4],[2,5],[3,7],[6,8]]
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,4],[2,5],[3,7],[6,8]]
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,4],[2,5],[3,6],[7,8]]
=> 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,4],[2,5],[3,6],[7,8]]
=> 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,4],[2,5],[3,6],[7,8]]
=> 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,4],[2,5],[3,6],[7,8]]
=> 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,3],[2,6],[4,7],[5,8]]
=> 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> 3
Description
The Shynar inversion number of a standard tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000067
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,1] => [1,1,0,0]
=> [1,2] => [[1,0],[0,1]]
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => [[0,1],[1,0]]
=> 1
[2,1] => [1,0,1,0]
=> [2,1] => [[0,1],[1,0]]
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
Description
The inversion number of the alternating sign matrix.
If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as
$$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$
When restricted to permutation matrices, this gives the usual inversion number of the permutation.
Matching statistic: St000081
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,1] => [1,1,0,0]
=> [1,2] => ([],2)
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[2,1] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
Description
The number of edges of a graph.
The following 90 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. St000332The positive inversions of an alternating sign matrix. St001034The area of the parallelogram polyomino associated with the Dyck path. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000795The mad of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001214The aft of an integer partition. St001176The size of a partition minus its first part. St000507The number of ascents of a standard tableau. St000010The length of the partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000288The number of ones in a binary word. St000820The number of compositions obtained by rotating the composition. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000097The order of the largest clique of the graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001581The achromatic number of a graph. St000147The largest part of an integer partition. St000098The chromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000011The number of touch points (or returns) of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules 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:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000306The bounce count of a Dyck path. St001875The number of simple modules with projective dimension at most 1. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000553The number of blocks of a graph. St000822The Hadwiger number of the graph. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000222The number of alignments in the permutation. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001535The number of cyclic alignments of a permutation. St001841The number of inversions of a set partition. St001911A descent variant minus the number of inversions. St000570The Edelman-Greene number of a permutation.
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