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Your data matches 140 different statistics following compositions of up to 3 maps.
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Matching statistic: St000454
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ 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
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 0
[2]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,3,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ([(5,6)],7)
=> 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001165
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 3 + 1
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001394
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,2] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,6,7,2,4,5] => ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2,7,8] => ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [3,4,1,7,8,2,5,6] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,2,6,7,3,4,5] => 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [3,1,2,7,8,4,5,6] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6,8] => ? = 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,2,3,7,8,4,5,6] => ? = 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4,8] => 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,8,2,4,5,6,7] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8] => ? = 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => 0
Description
The genus of a permutation.
The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation
$$
n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),
$$
where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.
Matching statistic: St001198
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1]
=> [1,0]
=> ? = 0
[2,1]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5]
=> []
=> []
=> []
=> ? = 2
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 1
[5,1]
=> [1]
=> [1]
=> [1,0]
=> ? = 2
[2,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? = 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 2
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[3,3,3]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[5,5]
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[5,3,2]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,3]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[3,3,3,2]
=> [3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,3,3,1,1]
=> [3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[5,5,2]
=> [5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[4,3,3,2]
=> [3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1]
=> [1,0]
=> ? = 0
[2,1]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5]
=> []
=> []
=> []
=> ? = 2
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 1
[5,1]
=> [1]
=> [1]
=> [1,0]
=> ? = 2
[2,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? = 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 2
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[3,3,3]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[5,5]
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[5,3,2]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,3]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[3,3,3,2]
=> [3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,3,3,1,1]
=> [3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[5,5,2]
=> [5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[4,3,3,2]
=> [3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000836
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 75%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> [2,1] => [1,2] => 0
[2]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,1,3,2] => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [3,4,5,6,1,2,7] => ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [5,2,6,1,4,3] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,2,4,5,6] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [3,6,2,7,1,5,4] => ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [3,4,5,6,1,2,7,8] => ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [3,4,7,1,5,6,2] => ? = 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [3,4,7,2,8,1,6,5] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6,1,3,4,5,2] => 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [6,2,3,7,1,5,4] => ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [3,7,2,4,8,1,6,5] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [5,6,7,1,4,2,3] => ? = 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [3,4,7,1,5,6,2,8] => ? = 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,2,4,5,6,7] => ? = 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [7,2,1,4,5,6,3] => ? = 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => [7,2,3,4,8,1,6,5] => ? = 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,7,5,1,2,3,8,4] => [5,6,7,1,4,2,3,8] => ? = 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,8,2,1,5,6,7,4] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,2,4,5,6,7,8] => ? = 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => [8,2,3,1,5,6,7,4] => ? = 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => ? = 0
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St001142
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 75%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> ? = 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001569
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => ? = 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,1,3,4] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => ? = 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => ? = 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? = 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => ? = 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => ? = 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ? = 0
Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St000291
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 75%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> 10 => 11 => 0
[2]
=> [1,0,1,0]
=> 1010 => 1101 => 1
[1,1]
=> [1,1,0,0]
=> 1100 => 0111 => 0
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => 110011 => 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => 001111 => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 11101001 => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1101010101 => ? = 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 11000111 => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 110101010011 => ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 00011111 => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0111101001 => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 1100001111 => ? = 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => 110011101001 => ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => 11010101000111 => ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 110101110001 => ? = 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => 11010011101001 => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1111100001 => ? = 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => 001111101001 => ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0000111111 => 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> 10111100010100 => 11000111101001 => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => 111101010001 => ? = 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => 11010111000011 => ? = 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => 110000011111 => ? = 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 111110010000 => 011111100001 => ? = 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 11111000010100 => 00011111101001 => ? = 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 11101010110000 => 11110101000011 => ? = 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 10111110010000 => 11001111100001 => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => 000001111111 => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => 11000000111111 => ? = 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 11111100010000 => 00111111100001 => ? = 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => 00000011111111 => ? = 0
Description
The number of descents of a binary word.
Matching statistic: St001769
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 75%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => ? = 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => [2,1,5,6,3,4] => ? = 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => [2,3,4,5,1,6,7] => ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? = 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [2,3,1,6,7,4,5] => ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,3,6,7,4,5] => [2,1,3,6,7,4,5] => ? = 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [2,3,6,1,4,5,7] => ? = 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 2
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ? = 2
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 3
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
Description
The reflection length of a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
The following 130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001864The number of excedances of a signed permutation. St000390The number of runs of ones in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001712The number of natural descents of a standard Young tableau. St000005The bounce statistic of a Dyck path. St000260The radius of a connected graph. St000331The number of upper interactions of a Dyck path. St000358The number of occurrences of the pattern 31-2. St000711The number of big exceedences of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001728The number of invisible descents of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000015The number of peaks of a Dyck path. 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. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000702The number of weak deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000740The last entry of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000942The number of critical left to right maxima of the parking functions. St000991The number of right-to-left minima of a permutation. 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 $eA$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St001665The number of pure excedances of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000456The monochromatic index of a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000264The girth of a graph, which is not a tree. St000903The number of different parts of an integer composition. St000023The number of inner peaks of a permutation. St000077The number of boxed and circled entries. St000232The number of crossings of a set partition. St000234The number of global ascents of a permutation. St000353The number of inner valleys of a permutation. St000472The sum of the ascent bottoms of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000497The lcb statistic of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000732The number of double deficiencies of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000502The number of successions of a set partitions. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001557The number of inversions of the second entry of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001937The size of the center of a parking function. St000824The sum of the number of descents and the number of recoils of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001520The number of strict 3-descents. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000632The jump number of the poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001722The number of minimal chains with small intervals between a binary word and the top element. St000307The number of rowmotion orbits of a poset. St001060The distinguishing index of a graph. St000717The number of ordinal summands of a poset.
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