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Your data matches 93 different statistics following compositions of up to 3 maps.
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Matching statistic: St000719
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000719: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 0
[(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> 0
[(1,4),(2,3)]
=> 0
[(1,2),(3,4),(5,6)]
=> 3
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 0
[(1,5),(2,4),(3,6)]
=> 0
[(1,4),(2,5),(3,6)]
=> 0
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 2
[(1,2),(3,6),(4,5)]
=> 2
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 0
[(1,5),(2,6),(3,4)]
=> 0
[(1,6),(2,5),(3,4)]
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> 6
[(1,3),(2,4),(5,6),(7,8)]
=> 5
[(1,4),(2,3),(5,6),(7,8)]
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> 4
[(1,6),(2,3),(4,5),(7,8)]
=> 4
[(1,7),(2,3),(4,5),(6,8)]
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> 4
[(1,2),(3,5),(4,6),(7,8)]
=> 5
[(1,2),(3,6),(4,5),(7,8)]
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> 4
[(1,4),(2,6),(3,5),(7,8)]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> 4
[(1,2),(3,8),(4,5),(6,7)]
=> 4
[(1,3),(2,8),(4,5),(6,7)]
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> 2
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: St000018
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> []
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000246
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
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: St001759
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
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
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata 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
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2
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
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> [[1],[2]]
=> [[1,2]]
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3],[2,4],[5,6]]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,3],[2,4],[5,6]]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2],[3,5],[4,6]]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,3],[2,5],[4,6]]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,3],[2,4],[5,6],[7,8]]
=> 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,3],[2,4],[5,6],[7,8]]
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> 4
[(1,7),(2,3),(4,5),(6,8)]
=> [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
[(1,8),(2,3),(4,5),(6,7)]
=> [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
[(1,8),(2,4),(3,5),(6,7)]
=> [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
[(1,7),(2,4),(3,5),(6,8)]
=> [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
[(1,6),(2,4),(3,5),(7,8)]
=> [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
[(1,5),(2,4),(3,6),(7,8)]
=> [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),(2,5),(3,6),(7,8)]
=> [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
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2],[3,5],[4,6],[7,8]]
=> 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2],[3,5],[4,6],[7,8]]
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,3],[2,5],[4,6],[7,8]]
=> 4
[(1,4),(2,6),(3,5),(7,8)]
=> [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
[(1,5),(2,6),(3,4),(7,8)]
=> [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
[(1,6),(2,5),(3,4),(7,8)]
=> [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
[(1,7),(2,5),(3,4),(6,8)]
=> [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
[(1,8),(2,5),(3,4),(6,7)]
=> [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
[(1,8),(2,6),(3,4),(5,7)]
=> [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
[(1,7),(2,6),(3,4),(5,8)]
=> [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
[(1,6),(2,7),(3,4),(5,8)]
=> [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
[(1,5),(2,7),(3,4),(6,8)]
=> [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
[(1,4),(2,7),(3,5),(6,8)]
=> [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),(2,7),(4,5),(6,8)]
=> [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),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2],[3,5],[4,7],[6,8]]
=> 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2],[3,5],[4,7],[6,8]]
=> 4
[(1,3),(2,8),(4,5),(6,7)]
=> [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
[(1,4),(2,8),(3,5),(6,7)]
=> [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
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
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> [1] => [[1]]
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => [[0,1],[1,0]]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => [[1,0],[0,1]]
=> 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => [[1,0],[0,1]]
=> 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[(1,7),(2,3),(4,5),(6,8)]
=> [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
[(1,8),(2,3),(4,5),(6,7)]
=> [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
[(1,8),(2,4),(3,5),(6,7)]
=> [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
[(1,7),(2,4),(3,5),(6,8)]
=> [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
[(1,6),(2,4),(3,5),(7,8)]
=> [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
[(1,5),(2,4),(3,6),(7,8)]
=> [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),(2,5),(3,6),(7,8)]
=> [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
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[(1,4),(2,6),(3,5),(7,8)]
=> [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
[(1,5),(2,6),(3,4),(7,8)]
=> [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
[(1,6),(2,5),(3,4),(7,8)]
=> [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
[(1,7),(2,5),(3,4),(6,8)]
=> [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
[(1,8),(2,5),(3,4),(6,7)]
=> [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
[(1,8),(2,6),(3,4),(5,7)]
=> [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
[(1,7),(2,6),(3,4),(5,8)]
=> [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
[(1,6),(2,7),(3,4),(5,8)]
=> [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
[(1,5),(2,7),(3,4),(6,8)]
=> [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
[(1,4),(2,7),(3,5),(6,8)]
=> [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),(2,7),(4,5),(6,8)]
=> [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),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[(1,3),(2,8),(4,5),(6,7)]
=> [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
[(1,4),(2,8),(3,5),(6,7)]
=> [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
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
Mp00150: Perfect matchings —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
[(1,2)]
=> [1,0]
=> [1] => ([],1)
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
Description
The number of edges of a graph.
Matching statistic: St000293
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> 10 => 01 => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> 1010 => 0101 => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> 1100 => 0011 => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> 1100 => 0011 => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 101010 => 010101 => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 110010 => 010011 => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 110010 => 010011 => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> 110100 => 001011 => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 110100 => 001011 => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> 111000 => 000111 => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> 111000 => 000111 => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> 111000 => 000111 => 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> 110100 => 001011 => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> 101100 => 001101 => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> 101100 => 001101 => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> 110100 => 001011 => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> 111000 => 000111 => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> 111000 => 000111 => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 111000 => 000111 => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 6
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01010011 => 5
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 01010011 => 5
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01001011 => 4
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01001011 => 4
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01001011 => 4
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 01001101 => 5
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 01001101 => 5
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 01001011 => 4
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01000111 => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 00010111 => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00101101 => 4
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 00101101 => 4
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 00101011 => 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 00100111 => 2
Description
The number of inversions of a binary word.
The following 83 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. 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. St000264The girth of a graph, which is not a tree. St000422The energy of a graph, if it is integral. St000834The number of right outer peaks of a permutation. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St001052The length of the exterior of a permutation. St000497The lcb statistic of a set partition. St000366The number of double descents of a permutation. St000232The number of crossings of a set partition. 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. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000352The Elizalde-Pak rank of a permutation. St000883The number of longest increasing subsequences of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001096The size of the overlap set of a permutation. St000007The number of saliances of the permutation. St000842The breadth of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000233The number of nestings of a set partition. St000356The number of occurrences of the pattern 13-2. St000731The number of double exceedences of a permutation. St001330The hat guessing number of a graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000496The rcs statistic of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000028The number of stack-sorts needed to sort a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000441The number of successions of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000022The number of fixed points of a permutation.
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