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Your data matches 198 different statistics following compositions of up to 3 maps.
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Matching statistic: St000066
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> 1
[[0,1],[1,0]]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 3
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Matching statistic: St000193
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St000193: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> 1
[[0,1],[1,0]]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> 2
[[0,1,0],[0,0,1],[1,0,0]]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 4
Description
The row of the unique '1' in the first column of the alternating sign matrix.
Matching statistic: St000199
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000199: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> 2
[[0,1],[1,0]]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> 2
[[0,1,0],[0,0,1],[1,0,0]]
=> 1
[[0,0,1],[0,1,0],[1,0,0]]
=> 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 1
Description
The column of the unique '1' in the last row of the alternating sign matrix.
Matching statistic: St000200
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
St000200: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> 2
[[0,1],[1,0]]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 2
Description
The row of the unique '1' in the last column of the alternating sign matrix.
Matching statistic: St000898
St000898: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> 1
[[0,1],[1,0]]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> 2
[[0,1,0],[1,0,0],[0,0,1]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> 2
[[0,1,0],[0,0,1],[1,0,0]]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 3
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 3
Description
The number of maximal entries in the last diagonal of the monotone triangle.
Consider the alternating sign matrix
$$
\left(\begin{array}{rrrrr}
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0
\end{array}\right).
$$
The corresponding monotone triangle is
$$
\begin{array}{ccccccccc}
5 & & 4 & & 3 & & 2 & & 1 \\
& 5 & & 4 & & 3 & & 1 & \\
& & 5 & & 3 & & 1 & & \\
& & & 5 & & 3 & & & \\
& & & & 4 & & & &
\end{array}
$$
The first entry $1$ in the last diagonal is maximal, because rows are strictly decreasing and its left neighbour is $2$. Also, the entry $3$ in the last diagonal is maximal, because diagonals from north-west to south-east are weakly decreasing, and its north-west neighbour is also $3$. All other entries in the last diagonal are non-maximal, thus the statistic on this matrix is $2$.
Conjecturally, this statistic is equidistributed with [[St000066]].
Matching statistic: St000025
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [1,0,1,0]
=> 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000740
(load all 88 compositions to match this statistic)
(load all 88 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [1,2] => 2
[[0,1],[1,0]]
=> [2,1] => 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 2
[[0,1,0],[0,0,1],[1,0,0]]
=> [2,3,1] => 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [2,3,1,4] => 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,3,4,2] => 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [3,2,4,1] => 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000746
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
St000746: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000746: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> 1
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> 2
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3
Description
The number of pairs with odd minimum in a perfect matching.
Matching statistic: St001184
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [1,0,1,0]
=> 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 3
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001291
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0,1]]
=> [1,0,1,0]
=> 2
[[0,1],[1,0]]
=> [1,1,0,0]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> 2
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
The following 188 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000054The first entry of the permutation. St000061The number of nodes on the left branch of a binary tree. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001497The position of the largest weak excedence of a permutation. St000010The length of the partition. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000738The first entry in the last row of a standard tableau. St000840The number of closers smaller than the largest opener in a perfect matching. St000989The number of final rises of a permutation. St001153The number of blocks with even minimum in a set partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000240The number of indices that are not small excedances. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000617The number of global maxima of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000702The number of weak deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000839The largest opener of a set partition. St000843The decomposition number of a perfect matching. St000990The first ascent of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000067The inversion number of the alternating sign matrix. St000083The number of left oriented leafs of a binary tree except the first one. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000203The number of external nodes of a binary tree. St000204The number of internal nodes of a binary tree. St000216The absolute length of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000304The load of a permutation. St000331The number of upper interactions of a Dyck path. St000356The number of occurrences of the pattern 13-2. St000378The diagonal inversion number of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000502The number of successions of a set partitions. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000993The multiplicity of the largest part of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St001115The number of even descents of a permutation. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph. St000648The number of 2-excedences of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000153The number of adjacent cycles of a permutation. St000871The number of very big ascents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000662The staircase size of the code of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000665The number of rafts of a permutation. St001096The size of the overlap set of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001581The achromatic number of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001394The genus of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000022The number of fixed points of a permutation. St000451The length of the longest pattern of the form k 1 2. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000918The 2-limited packing number of a graph. St001116The game chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000351The determinant of the adjacency matrix of a graph. St000456The monochromatic index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000115The single entry in the last row. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000942The number of critical left to right maxima of the parking functions. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000005The bounce statistic of a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001118The acyclic chromatic index of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001645The pebbling number of a connected graph. St001812The biclique partition number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000306The bounce count of a Dyck path. St000386The number of factors DDU in a Dyck path. St000445The number of rises of length 1 of a Dyck path. St001052The length of the exterior of a permutation.
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