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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St000653
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => 3
[1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => [3,1,4,2] => 3
[2,4,3,1] => [4,1,3,2] => 3
[3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,1,3] => 2
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [4,3,1,2] => 2
[4,1,2,3] => [2,3,4,1] => 3
[4,1,3,2] => [2,4,3,1] => 3
[4,2,1,3] => [3,2,4,1] => 3
[4,2,3,1] => [4,2,3,1] => 3
[4,3,1,2] => [3,4,2,1] => 3
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => [1,2,5,3,4] => 3
[1,2,5,3,4] => [1,2,4,5,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => 2
[1,3,5,2,4] => [1,4,2,5,3] => 4
[1,3,5,4,2] => [1,5,2,4,3] => 4
[1,4,2,3,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,3,5,2,4] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 3
[1,4,3,5,2] => [1,5,3,2,4] => 3
[1,4,5,2,3] => [1,4,5,2,3] => 3
[1,4,5,3,2] => [1,5,4,2,3] => 3
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000727
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => 2 = 1 + 1
[1,2,3] => [3,2,1] => 1 = 0 + 1
[1,3,2] => [2,3,1] => 3 = 2 + 1
[2,1,3] => [3,1,2] => 2 = 1 + 1
[2,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [2,1,3] => 3 = 2 + 1
[3,2,1] => [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => 4 = 3 + 1
[1,3,2,4] => [4,2,3,1] => 3 = 2 + 1
[1,3,4,2] => [2,4,3,1] => 3 = 2 + 1
[1,4,2,3] => [3,2,4,1] => 4 = 3 + 1
[1,4,3,2] => [2,3,4,1] => 4 = 3 + 1
[2,1,3,4] => [4,3,1,2] => 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => 4 = 3 + 1
[2,3,1,4] => [4,1,3,2] => 2 = 1 + 1
[2,3,4,1] => [1,4,3,2] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => 4 = 3 + 1
[2,4,3,1] => [1,3,4,2] => 4 = 3 + 1
[3,1,2,4] => [4,2,1,3] => 3 = 2 + 1
[3,1,4,2] => [2,4,1,3] => 3 = 2 + 1
[3,2,1,4] => [4,1,2,3] => 3 = 2 + 1
[3,2,4,1] => [1,4,2,3] => 3 = 2 + 1
[3,4,1,2] => [2,1,4,3] => 3 = 2 + 1
[3,4,2,1] => [1,2,4,3] => 3 = 2 + 1
[4,1,2,3] => [3,2,1,4] => 4 = 3 + 1
[4,1,3,2] => [2,3,1,4] => 4 = 3 + 1
[4,2,1,3] => [3,1,2,4] => 4 = 3 + 1
[4,2,3,1] => [1,3,2,4] => 4 = 3 + 1
[4,3,1,2] => [2,1,3,4] => 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => 5 = 4 + 1
[1,2,4,3,5] => [5,3,4,2,1] => 4 = 3 + 1
[1,2,4,5,3] => [3,5,4,2,1] => 4 = 3 + 1
[1,2,5,3,4] => [4,3,5,2,1] => 5 = 4 + 1
[1,2,5,4,3] => [3,4,5,2,1] => 5 = 4 + 1
[1,3,2,4,5] => [5,4,2,3,1] => 3 = 2 + 1
[1,3,2,5,4] => [4,5,2,3,1] => 5 = 4 + 1
[1,3,4,2,5] => [5,2,4,3,1] => 3 = 2 + 1
[1,3,4,5,2] => [2,5,4,3,1] => 3 = 2 + 1
[1,3,5,2,4] => [4,2,5,3,1] => 5 = 4 + 1
[1,3,5,4,2] => [2,4,5,3,1] => 5 = 4 + 1
[1,4,2,3,5] => [5,3,2,4,1] => 4 = 3 + 1
[1,4,2,5,3] => [3,5,2,4,1] => 4 = 3 + 1
[1,4,3,2,5] => [5,2,3,4,1] => 4 = 3 + 1
[1,4,3,5,2] => [2,5,3,4,1] => 4 = 3 + 1
[1,4,5,2,3] => [3,2,5,4,1] => 4 = 3 + 1
[1,4,5,3,2] => [2,3,5,4,1] => 4 = 3 + 1
Description
The largest label of a leaf in the binary search tree associated with the permutation.
Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
Matching statistic: St001480
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St000740
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4 = 3 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4 = 3 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4 = 3 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4 = 3 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 5 = 4 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 4 = 3 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4 = 3 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 5 = 4 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 5 = 4 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 5 = 4 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 5 = 4 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 5 = 4 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 4 = 3 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 4 = 3 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4 = 3 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4 = 3 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001291
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
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 $A$.
Matching statistic: St000019
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,2] => 0
[2,1] => [[.,.],.]
=> [1,2] => [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1,3] => 1
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [2,1,3] => 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [3,2,1] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [2,3,1] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,2,3] => 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1,2,4] => 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1,2,4] => 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,1,2] => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,4,1,2] => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,3,4] => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,3,4] => 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,3,4] => 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,3,2,1] => 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,4,2,1] => 3
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,4,3,1] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,4,3,1] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,2,4,1] => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,2,3,5] => 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,2,3,5] => 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,1,2,3] => 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,5,1,2,3] => 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,2,4,5] => 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,2,4] => 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,2,4,5] => 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,2,4,5] => 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,2,4] => 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,2,4] => 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,2,5] => 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,2,5] => 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,4,1,2,5] => 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,4,1,2,5] => 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,2,5] => 3
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,4,1,2,5] => 3
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000067
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00035: Dyck paths —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%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00035: Dyck paths —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] => [1,0,1,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[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] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
Description
The inversion number of the alternating sign matrix.
If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as
$$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$
When restricted to permutation matrices, this gives the usual inversion number of the permutation.
Matching statistic: St000141
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [[.,.],.]
=> [1,2] => 0
[2,1] => [1,2] => [.,[.,.]]
=> [2,1] => 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,3,2] => [2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 2
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 2
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 4
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 4
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 4
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 4
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 4
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 4
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 3
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[1,4,5,3,2] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 3
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000147
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 4
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000171
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 1
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St000054The first entry of the permutation. St000240The number of indices that are not small excedances. St000501The size of the first part in the decomposition of a permutation. St000734The last entry in the first row of a standard tableau. St000839The largest opener of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St001497The position of the largest weak excedence of a permutation. St001725The harmonious chromatic number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 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. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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