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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000673
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 2
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 3
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 4
[4,1,2,3] => 4
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 2
[4,3,1,2] => 4
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 4
[1,3,4,2,5] => 3
[1,3,4,5,2] => 4
[1,3,5,2,4] => 4
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 4
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 4
[1,4,5,3,2] => 4
Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001279
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 2
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1]
=> 2
[2,3,1] => [3]
=> 3
[3,1,2] => [3]
=> 3
[3,2,1] => [2,1]
=> 2
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [3,1]
=> 3
[1,4,2,3] => [3,1]
=> 3
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> 4
[2,3,1,4] => [3,1]
=> 3
[2,3,4,1] => [4]
=> 4
[2,4,1,3] => [4]
=> 4
[2,4,3,1] => [3,1]
=> 3
[3,1,2,4] => [3,1]
=> 3
[3,1,4,2] => [4]
=> 4
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [3,1]
=> 3
[3,4,1,2] => [2,2]
=> 4
[3,4,2,1] => [4]
=> 4
[4,1,2,3] => [4]
=> 4
[4,1,3,2] => [3,1]
=> 3
[4,2,1,3] => [3,1]
=> 3
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [4]
=> 4
[4,3,2,1] => [2,2]
=> 4
[1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> 3
[1,2,5,3,4] => [3,1,1]
=> 3
[1,2,5,4,3] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> 4
[1,3,4,2,5] => [3,1,1]
=> 3
[1,3,4,5,2] => [4,1]
=> 4
[1,3,5,2,4] => [4,1]
=> 4
[1,3,5,4,2] => [3,1,1]
=> 3
[1,4,2,3,5] => [3,1,1]
=> 3
[1,4,2,5,3] => [4,1]
=> 4
[1,4,3,2,5] => [2,1,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 3
[1,4,5,2,3] => [2,2,1]
=> 4
[1,4,5,3,2] => [4,1]
=> 4
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000235
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
St000235: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 2
[1,2,3] => [2,3,1] => 0
[1,3,2] => [3,2,1] => 2
[2,1,3] => [1,3,2] => 2
[2,3,1] => [1,2,3] => 3
[3,1,2] => [3,1,2] => 3
[3,2,1] => [2,1,3] => 2
[1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [2,4,3,1] => 2
[1,3,2,4] => [3,2,4,1] => 2
[1,3,4,2] => [4,2,3,1] => 3
[1,4,2,3] => [3,4,2,1] => 3
[1,4,3,2] => [4,3,2,1] => 2
[2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [1,4,3,2] => 4
[2,3,1,4] => [1,2,4,3] => 3
[2,3,4,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,4,2,3] => 4
[2,4,3,1] => [1,3,2,4] => 3
[3,1,2,4] => [3,1,4,2] => 3
[3,1,4,2] => [4,1,3,2] => 4
[3,2,1,4] => [2,1,4,3] => 2
[3,2,4,1] => [2,1,3,4] => 3
[3,4,1,2] => [4,1,2,3] => 4
[3,4,2,1] => [3,1,2,4] => 4
[4,1,2,3] => [3,4,1,2] => 4
[4,1,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [2,4,1,3] => 3
[4,2,3,1] => [2,3,1,4] => 2
[4,3,1,2] => [4,2,1,3] => 4
[4,3,2,1] => [3,2,1,4] => 4
[1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [2,3,5,4,1] => 2
[1,2,4,3,5] => [2,4,3,5,1] => 2
[1,2,4,5,3] => [2,5,3,4,1] => 3
[1,2,5,3,4] => [2,4,5,3,1] => 3
[1,2,5,4,3] => [2,5,4,3,1] => 2
[1,3,2,4,5] => [3,2,4,5,1] => 2
[1,3,2,5,4] => [3,2,5,4,1] => 4
[1,3,4,2,5] => [4,2,3,5,1] => 3
[1,3,4,5,2] => [5,2,3,4,1] => 4
[1,3,5,2,4] => [4,2,5,3,1] => 4
[1,3,5,4,2] => [5,2,4,3,1] => 3
[1,4,2,3,5] => [3,4,2,5,1] => 3
[1,4,2,5,3] => [3,5,2,4,1] => 4
[1,4,3,2,5] => [4,3,2,5,1] => 2
[1,4,3,5,2] => [5,3,2,4,1] => 3
[1,4,5,2,3] => [4,5,2,3,1] => 4
[1,4,5,3,2] => [5,4,2,3,1] => 4
[1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 2
[1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 2
[1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => ? = 3
[1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 3
[1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 2
[1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 2
[1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 4
[1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => ? = 3
[1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => ? = 4
[1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 4
[1,2,3,5,7,6,4] => [2,3,7,4,6,5,1] => ? = 3
[1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => ? = 3
[1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 4
[1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 2
[1,2,3,6,5,7,4] => [2,3,7,5,4,6,1] => ? = 3
[1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => ? = 4
[1,2,3,6,7,5,4] => [2,3,7,6,4,5,1] => ? = 4
[1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 4
[1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => ? = 3
[1,2,3,7,5,4,6] => [2,3,6,5,7,4,1] => ? = 3
[1,2,3,7,5,6,4] => [2,3,7,5,6,4,1] => ? = 2
[1,2,3,7,6,4,5] => [2,3,6,7,5,4,1] => ? = 4
[1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 4
[1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 2
[1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 4
[1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 4
[1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => ? = 5
[1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 5
[1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 4
[1,2,4,5,3,6,7] => [2,5,3,4,6,7,1] => ? = 3
[1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => ? = 5
[1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => ? = 4
[1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => ? = 5
[1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 5
[1,2,4,5,7,6,3] => [2,7,3,4,6,5,1] => ? = 4
[1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => ? = 4
[1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => ? = 5
[1,2,4,6,5,3,7] => [2,6,3,5,4,7,1] => ? = 3
[1,2,4,6,5,7,3] => [2,7,3,5,4,6,1] => ? = 4
[1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => ? = 5
[1,2,4,6,7,5,3] => [2,7,3,6,4,5,1] => ? = 5
[1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => ? = 5
[1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => ? = 4
[1,2,4,7,5,3,6] => [2,6,3,5,7,4,1] => ? = 4
[1,2,4,7,5,6,3] => [2,7,3,5,6,4,1] => ? = 3
[1,2,4,7,6,3,5] => [2,6,3,7,5,4,1] => ? = 5
[1,2,4,7,6,5,3] => [2,7,3,6,5,4,1] => ? = 5
[1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => ? = 3
[1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => ? = 5
Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St000896
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
St000896: Alternating sign matrices ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,0],[0,1]]
=> 0
[2,1] => [[0,1],[1,0]]
=> 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 3
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 2
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 4
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 3
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 4
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 4
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 4
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 3
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 3
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 4
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 4
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 4
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 4
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 3
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 4
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 4
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3
[1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3
[1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
[1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4
[1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3
[1,2,5,3,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[1,2,5,4,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 3
[1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 4
[1,2,5,6,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 4
[1,2,6,3,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 4
[1,2,6,3,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 3
[1,2,6,4,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 3
[1,2,6,4,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,2,6,5,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 4
[1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 4
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 5
[1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 4
[1,3,4,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,3,4,6,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 5
[1,3,4,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,3,5,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 5
[1,3,5,6,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 5
[1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 5
[1,3,6,2,5,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
[1,3,6,4,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 4
[1,3,6,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 3
[1,3,6,5,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
[1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
[1,4,2,5,6,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 5
[1,4,2,6,5,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 4
[1,4,3,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
[1,4,3,6,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4
[1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,4,5,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 5
[1,4,5,6,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 5
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 5
[1,4,6,2,5,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
[1,4,6,3,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 5
[1,4,6,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
[1,4,6,5,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
[1,4,6,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
[1,5,2,3,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 5
[1,5,2,4,6,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
Description
The number of zeros on the main diagonal of an alternating sign matrix.
Matching statistic: St001604
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%
Values
[1,2] => [1,2] => [-1,-2] => []
=> ? = 0 - 3
[2,1] => [2,1] => [-2,-1] => [2]
=> ? = 2 - 3
[1,2,3] => [1,2,3] => [-1,-2,-3] => []
=> ? = 0 - 3
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> ? = 2 - 3
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> ? = 2 - 3
[2,3,1] => [2,3,1] => [-2,-3,-1] => []
=> ? = 3 - 3
[3,1,2] => [3,1,2] => [-3,-1,-2] => []
=> ? = 3 - 3
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> ? = 2 - 3
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => []
=> ? = 0 - 3
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> ? = 2 - 3
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> ? = 2 - 3
[1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => []
=> ? = 3 - 3
[1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => []
=> ? = 3 - 3
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> ? = 2 - 3
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> ? = 2 - 3
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 1 = 4 - 3
[2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => []
=> ? = 3 - 3
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 1 = 4 - 3
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 1 = 4 - 3
[2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => []
=> ? = 3 - 3
[3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => []
=> ? = 3 - 3
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 1 = 4 - 3
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> ? = 2 - 3
[3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => []
=> ? = 3 - 3
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 1 = 4 - 3
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 1 = 4 - 3
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 1 = 4 - 3
[4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => []
=> ? = 3 - 3
[4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => []
=> ? = 3 - 3
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> ? = 2 - 3
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 1 = 4 - 3
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 1 = 4 - 3
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => []
=> ? = 0 - 3
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> ? = 2 - 3
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> ? = 2 - 3
[1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => []
=> ? = 3 - 3
[1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => []
=> ? = 3 - 3
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> ? = 2 - 3
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> ? = 2 - 3
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 1 = 4 - 3
[1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => []
=> ? = 3 - 3
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 1 = 4 - 3
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 1 = 4 - 3
[1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => []
=> ? = 3 - 3
[1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => []
=> ? = 3 - 3
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 1 = 4 - 3
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> ? = 2 - 3
[1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => []
=> ? = 3 - 3
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 1 = 4 - 3
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 1 = 4 - 3
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 1 = 4 - 3
[1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => []
=> ? = 3 - 3
[1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => []
=> ? = 3 - 3
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> ? = 2 - 3
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 1 = 4 - 3
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 1 = 4 - 3
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> ? = 2 - 3
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 1 = 4 - 3
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 1 = 4 - 3
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> ? = 5 - 3
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> ? = 5 - 3
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 1 = 4 - 3
[2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => []
=> ? = 3 - 3
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> ? = 5 - 3
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 1 = 4 - 3
[2,3,4,5,1] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => []
=> ? = 5 - 3
[2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => []
=> ? = 5 - 3
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 1 = 4 - 3
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 1 = 4 - 3
[2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => []
=> ? = 5 - 3
[2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => []
=> ? = 3 - 3
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 1 = 4 - 3
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> ? = 5 - 3
[2,4,5,3,1] => [2,4,5,3,1] => [-2,-4,-5,-3,-1] => []
=> ? = 5 - 3
[2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => []
=> ? = 5 - 3
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 1 = 4 - 3
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 1 = 4 - 3
[3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => [4]
=> 1 = 4 - 3
[3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => [4]
=> 1 = 4 - 3
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [2,2]
=> 1 = 4 - 3
[3,2,4,5,1] => [3,2,4,5,1] => [-3,-2,-4,-5,-1] => [4]
=> 1 = 4 - 3
[3,2,5,1,4] => [3,2,5,1,4] => [-3,-2,-5,-1,-4] => [4]
=> 1 = 4 - 3
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [2,2]
=> 1 = 4 - 3
[3,4,2,1,5] => [3,4,2,1,5] => [-3,-4,-2,-1,-5] => [4]
=> 1 = 4 - 3
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [2,2]
=> 1 = 4 - 3
[3,5,2,4,1] => [3,5,2,4,1] => [-3,-5,-2,-4,-1] => [4]
=> 1 = 4 - 3
[4,1,2,3,5] => [4,1,2,3,5] => [-4,-1,-2,-3,-5] => [4]
=> 1 = 4 - 3
[4,1,3,5,2] => [4,1,3,5,2] => [-4,-1,-3,-5,-2] => [4]
=> 1 = 4 - 3
[4,2,1,5,3] => [4,2,1,5,3] => [-4,-2,-1,-5,-3] => [4]
=> 1 = 4 - 3
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [2,2]
=> 1 = 4 - 3
[4,2,5,3,1] => [4,2,5,3,1] => [-4,-2,-5,-3,-1] => [4]
=> 1 = 4 - 3
[4,3,1,2,5] => [4,3,1,2,5] => [-4,-3,-1,-2,-5] => [4]
=> 1 = 4 - 3
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [2,2]
=> 1 = 4 - 3
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [2,2]
=> 1 = 4 - 3
[4,5,3,2,1] => [4,5,3,2,1] => [-4,-5,-3,-2,-1] => [4]
=> 1 = 4 - 3
[5,1,2,4,3] => [5,1,2,4,3] => [-5,-1,-2,-4,-3] => [4]
=> 1 = 4 - 3
[5,1,3,2,4] => [5,1,3,2,4] => [-5,-1,-3,-2,-4] => [4]
=> 1 = 4 - 3
[5,2,1,3,4] => [5,2,1,3,4] => [-5,-2,-1,-3,-4] => [4]
=> 1 = 4 - 3
[5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => [4]
=> 1 = 4 - 3
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [2,2]
=> 1 = 4 - 3
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
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