Processing math: 28%

Your data matches 157 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00160: Permutations graph of inversionsGraphs
Mp00203: Graphs coneGraphs
St001645: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([(0,1)],2)
=> 2
[1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 4
[2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7
[3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
Description
The pebbling number of a connected graph.
Matching statistic: St000300
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00160: Permutations graph of inversionsGraphs
St000300: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 2
[1,2] => [1,2] => [1,2] => ([],2)
=> 4
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 3
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,4,5,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,5,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,1,5,2] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,2,5,1] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,5,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,5,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,1,6,4,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,4,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,4,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
Description
The number of independent sets of vertices of a graph. An independent set of vertices of a graph G is a subset UV(G) such that no two vertices in U are adjacent. This is also the number of vertex covers of G as the map UV(G)U is a bijection between independent sets of vertices and vertex covers. The size of the largest independent set, also called independence number of G, is [[St000093]]
Matching statistic: St000867
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1]
=> 1 = 2 - 1
[1,2] => ([],2)
=> [1,1]
=> [2]
=> 3 = 4 - 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 3 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
Description
The sum of the hook lengths in the first row of an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition. Put differently, for a partition of size n with first parth λ1, this is \binom{\lambda_1}{2} + n.
Mp00160: Permutations graph of inversionsGraphs
Mp00147: Graphs squareGraphs
St001391: Graphs ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => ([],1)
=> ([],1)
=> 0 = 2 - 2
[1,2] => ([],2)
=> ([],2)
=> 2 = 4 - 2
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 4 - 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 5 - 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 6 - 2
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 7 - 2
[] => ([],0)
=> ?
=> ? = 1 - 2
Description
The disjunction number of a graph. Let V_n be the power set of \{1,\dots,n\} and let E_n=\{(a,b)| a,b\in V_n, a\neq b, a\cap b=\emptyset\}. Then the disjunction number of a graph G is the smallest integer n such that (V_n, E_n) has an induced subgraph isomorphic to G.
Matching statistic: St000180
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00065: Permutations permutation posetPosets
St000180: Posets ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => ([],1)
=> 2
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 4
[2,1] => [2,1] => [2,1] => ([],2)
=> 3
[3,2,1] => [3,2,1] => [3,2,1] => ([],3)
=> 4
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 5
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 5
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 5
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 5
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,3,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,4,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,4,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> 6
[3,4,5,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,4,5,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,4,6,1,5,2] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,4,6,2,5,1] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,4,6,5,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,4,6,5,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,5,1,6,4,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,5,4,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[3,5,4,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([],6)
=> 7
[] => [] => [] => ([],0)
=> ? = 1
Description
The number of chains of a poset.
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00160: Permutations graph of inversionsGraphs
St000301: Graphs ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => ([],1)
=> 2
[1,2] => [1,2] => [1,2] => ([],2)
=> 4
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 3
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,4,5,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,5,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,1,5,2] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,2,5,1] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,5,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,4,6,5,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,1,6,4,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,4,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,5,4,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[] => [] => [] => ([],0)
=> ? = 1
Description
The number of facets of the stable set polytope of a graph. The stable set polytope of a graph G is the convex hull of the characteristic vectors of stable (or independent) sets of vertices of G inside \mathbb{R}^{V(G)}.
Matching statistic: St001213
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001213: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1,0]
=> [1] => [1,0]
=> 2
[1,2] => [1,0,1,0]
=> [1,2] => [1,0,1,0]
=> 4
[2,1] => [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,1,5,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,5,1,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,5,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,4,6,1,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,4,6,2,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[] => []
=> [] => []
=> ? = 1
Description
The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.
Matching statistic: St001259
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001259: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => [1,0]
=> [1] => [1,0]
=> 2
[1,2] => [1,0,1,0]
=> [1,2] => [1,0,1,0]
=> 4
[2,1] => [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,1,5,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,5,1,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,4,6,5,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,4,6,1,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,5,4,6,2,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[] => []
=> [] => []
=> ? = 1
Description
The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.
Matching statistic: St001348
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001348: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => ([],1)
=> [1]
=> [1,0,1,0]
=> 2
[1,2] => ([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> 4
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[] => ([],0)
=> []
=> []
=> ? = 1
Description
The bounce of the parallelogram polyomino associated with the Dyck path. A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St001504
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001504: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 100%distinct values known / distinct values provided: 86%
Values
[1] => ([],1)
=> [1]
=> [1,0,1,0]
=> 2
[1,2] => ([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> 4
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7
[] => ([],0)
=> []
=> []
=> ? = 1
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
The following 147 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001808The box weight or horizontal decoration of a Dyck path. St000271The chromatic index of a graph. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St000326The position of the first one in a binary word after appending a 1 at the end. St000288The number of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000877The depth of the binary word interpreted as a path. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001480The number of simple summands of the module J^2/J^3. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000438The position of the last up step in a Dyck path. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000654The first descent of a permutation. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000808The number of up steps of the associated bargraph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000990The first ascent of a permutation. St000216The absolute length of a permutation. St000297The number of leading ones in a binary word. St000354The number of recoils of a permutation. St000503The maximal difference between two elements in a common block. 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. St000795The mad of a 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. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001721The degree of a binary word. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000225Difference between largest and smallest parts in a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000806The semiperimeter of the associated bargraph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000890The number of nonzero entries in an alternating sign matrix. 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. St000296The length of the symmetric border of a binary word. St000393The number of strictly increasing runs in a binary word. St000627The exponent of a binary word. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001267The length of the Lyndon factorization of the binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000519The largest length of a factor maximising the subword complexity. St000836The number of descents of distance 2 of a permutation. St000656The number of cuts of a poset. St000520The number of patterns in a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000064The number of one-box pattern of a permutation. St000444The length of the maximal rise of a Dyck path. St000477The weight of a partition according to Alladi. St000625The sum of the minimal distances to a greater element. St000668The least common multiple of the parts of the partition. St000673The number of non-fixed points of a permutation. St000708The product of the parts of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000770The major index of an integer partition when read from bottom to top. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000060The greater neighbor of the maximum. St000327The number of cover relations in a poset. St000382The first part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000657The smallest part of an integer composition. St000681The Grundy value of Chomp on Ferrers diagrams. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000878The number of ones minus the number of zeros of a binary word. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000983The length of the longest alternating subword. St001052The length of the exterior of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000219The number of occurrences of the pattern 231 in a permutation. St000293The number of inversions of a binary word. St000538The number of even inversions of a permutation. St000691The number of changes of a binary word. St000837The number of ascents of distance 2 of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001118The acyclic chromatic index of a graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000454The largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000474Dyson's crank of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000699The toughness times the least common multiple of 1,. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000087The number of induced subgraphs. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000224The sorting index of a permutation. St001117The game chromatic index of a graph. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001834The number of non-isomorphic minors of a graph. St000841The largest opener of a perfect matching. St001770The number of facets of a certain subword complex associated with the signed permutation. St001948The number of augmented double ascents of a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000906The length of the shortest maximal chain in a 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. St001927Sparre Andersen's number of positives of a signed permutation. 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). St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001093The detour number of a graph. St001512The minimum rank of a graph. St001545The second Elser number of a connected graph.