Processing math: 20%

Your data matches 26 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000147
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,2] => ([],2)
=> [1,1]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1]
=> 1
[1,2] => ([],2)
=> [1,1]
=> [2]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [3]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000734
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 90% values known / values provided: 100%distinct values known / distinct values provided: 90%
Values
[1] => ([],1)
=> [1]
=> [[1]]
=> 1
[1,2] => ([],2)
=> [1,1]
=> [[1],[2]]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[] => ([],0)
=> []
=> []
=> ? = 0
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000676: Dyck paths ⟶ ℤResult quality: 90% values known / values provided: 100%distinct values known / distinct values provided: 90%
Values
[1] => ([],1)
=> [1]
=> [1,0]
=> 1
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,3,4,5,6,7,8] => ([],8)
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,1,5,2,7,4,9,6,8] => ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[2,4,1,6,3,8,5,9,7] => ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,4,8,2,5,9,1,3,6,10] => ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[8,6,9,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[2,4,6,1,3,5,7,8,9] => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[7,9,5,8,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,5,2,6,3,7,8,9] => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,1,6,2,8,3,9,5,7] => ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,2,3,7,4,8,5,9,6] => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength n with k up steps in odd positions and k returns to the main diagonal are counted by the binomial coefficient \binom{n-1}{k-1} [3,4].
Matching statistic: St001039
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001039: Dyck paths ⟶ ℤResult quality: 80% values known / values provided: 100%distinct values known / distinct values provided: 80%
Values
[1] => ([],1)
=> [1]
=> [1,0]
=> ? = 1
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,3,4,5,6,7,8] => ([],8)
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,1,5,2,7,4,9,6,8] => ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[2,4,1,6,3,8,5,9,7] => ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[] => ([],0)
=> []
=> []
=> ? = 0
[7,4,8,2,5,9,1,3,6,10] => ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[8,6,9,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[2,4,6,1,3,5,7,8,9] => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[7,9,5,8,3,6,1,4,2] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,5,2,6,3,7,8,9] => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,1,6,2,8,3,9,5,7] => ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,2,3,7,4,8,5,9,6] => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00028: Dyck paths reverseDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
St000381: Integer compositions ⟶ ℤResult quality: 90% values known / values provided: 100%distinct values known / distinct values provided: 90%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4
[3,1,2,6,8,4,5,7] => ?
=> ?
=> ? => ? = 5
[3,1,5,2,7,4,9,6,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? => ? = 9
[] => []
=> []
=> [] => ? = 0
[7,4,8,2,5,9,1,3,6,10] => [1,1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? => ? = 9
[7,3,4,1,5,2,8,6] => ?
=> ?
=> ? => ? = 8
[6,2,3,1,7,4,8,5] => ?
=> ?
=> ? => ? = 8
[4,1,5,2,8,6,7,3] => ?
=> ?
=> ? => ? = 8
[8,4,5,1,6,2,7,3] => ?
=> ?
=> ? => ? = 8
[3,1,7,4,8,5,6,2] => ?
=> ?
=> ? => ? = 8
[8,7,1,6,2,4,5,3] => ?
=> ?
=> ? => ? = 8
[2,4,6,1,3,5,7,8,9] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? => ? = 6
[6,7,2,4,3,8,5,1] => ?
=> ?
=> ? => ? = 8
[4,1,5,2,6,3,7,8,9] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? => ? = 6
[1,8,2,3,7,5,6,4] => ?
=> ?
=> ? => ? = 7
[1,8,2,7,3,5,6,4] => ?
=> ?
=> ? => ? = 7
[7,1,6,2,4,5,3,8] => ?
=> ?
=> ? => ? = 7
[1,2,3,7,4,8,5,9,6] => [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? => ? = 6
[7,1,5,2,4,3,6,8] => ?
=> ?
=> ? => ? = 7
Description
The largest part of an integer composition.
Mp00159: Permutations Demazure product with inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000503: Set partitions ⟶ ℤResult quality: 80% values known / values provided: 94%distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0 = 1 - 1
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1 = 2 - 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3 = 4 - 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3 = 4 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[3,4,2,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[4,2,3,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7 - 1
[4,5,6,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,5,2,3,4,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,2,3,4,5,1,7,8] => [6,5,3,4,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[4,5,3,2,6,1,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[4,3,5,2,6,1,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[3,4,5,2,6,1,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[5,3,2,4,6,1,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[5,2,3,4,6,1,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[4,3,2,5,6,1,7,8] => [6,3,2,4,5,1,7,8] => ?
=> ?
=> ? = 6 - 1
[3,4,2,5,6,1,7,8] => [6,3,2,4,5,1,7,8] => ?
=> ?
=> ? = 6 - 1
[4,2,3,5,6,1,7,8] => [6,3,2,4,5,1,7,8] => ?
=> ?
=> ? = 6 - 1
[3,2,4,5,6,1,7,8] => [6,2,3,4,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[5,6,3,4,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[3,4,5,6,1,2,7,8] => [6,5,3,4,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,5,4,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[4,5,6,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,3,4,2,1,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,4,3,1,2,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,4,2,1,3,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,4,1,2,3,5,7,8] => [6,5,3,4,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,2,3,1,4,5,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[6,3,1,2,4,5,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6 - 1
[2,3,4,5,1,6,7,8] => [5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5 - 1
[5,1,2,3,4,6,7,8] => [5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5 - 1
[4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 4 - 1
[3,4,1,2,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 4 - 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,2,4,5,6,8,7,3] => [1,2,8,4,5,7,6,3] => ?
=> ?
=> ? = 6 - 1
[3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 5 - 1
[3,2,1,6,7,8,5,4] => [3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 5 - 1
[3,2,1,6,5,8,7,4] => [3,2,1,8,5,7,6,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 5 - 1
[2,3,1,6,5,8,7,4] => [3,2,1,8,5,7,6,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 5 - 1
[1,2,3,5,6,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 5 - 1
[4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[4,3,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[4,3,2,1,6,8,7,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[4,3,2,1,7,6,8,5] => [4,3,2,1,8,6,7,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[4,3,2,1,6,7,8,5] => [4,3,2,1,8,6,7,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[3,4,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[2,4,3,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[2,4,3,1,6,8,7,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
Description
The maximal difference between two elements in a common block.
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
St000141: Permutations ⟶ ℤResult quality: 90% values known / values provided: 93%distinct values known / distinct values provided: 90%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 3 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2 = 3 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 3 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 3 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 3 - 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 2 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 3 - 1
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,5,7,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 3 - 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 3 - 1
[1,2,3,6,5,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,7,4,5,6] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,7,4,6,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,7,5,4,6] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,7,6,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 2 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2 - 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3 - 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3 - 1
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 3 - 1
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 3 - 1
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 4 - 1
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,5,7,3,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 4 - 1
[1,2,4,6,3,7,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 4 - 1
[1,2,4,6,5,7,3] => [1,2,7,5,4,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,6,7,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,7,3,5,6] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,7,3,6,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,7,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,7,6,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,4,7,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 3 - 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 3 - 1
[1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 4 - 1
[1,2,5,3,6,7,4] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
[1,2,5,3,7,4,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5 - 1
Description
The maximum drop size of a permutation. The maximum drop size of a permutation \pi of [n]=\{1,2,\ldots, n\} is defined to be the maximum value of i-\pi(i).
Mp00114: Permutations connectivity setBinary words
Mp00105: Binary words complementBinary words
St000392: Binary words ⟶ ℤResult quality: 90% values known / values provided: 93%distinct values known / distinct values provided: 90%
Values
[1] => => => ? = 1 - 1
[1,2] => 1 => 0 => 0 = 1 - 1
[2,1] => 0 => 1 => 1 = 2 - 1
[1,2,3] => 11 => 00 => 0 = 1 - 1
[1,3,2] => 10 => 01 => 1 = 2 - 1
[2,1,3] => 01 => 10 => 1 = 2 - 1
[2,3,1] => 00 => 11 => 2 = 3 - 1
[3,1,2] => 00 => 11 => 2 = 3 - 1
[3,2,1] => 00 => 11 => 2 = 3 - 1
[1,2,3,4] => 111 => 000 => 0 = 1 - 1
[1,2,4,3] => 110 => 001 => 1 = 2 - 1
[1,3,2,4] => 101 => 010 => 1 = 2 - 1
[1,3,4,2] => 100 => 011 => 2 = 3 - 1
[1,4,2,3] => 100 => 011 => 2 = 3 - 1
[1,4,3,2] => 100 => 011 => 2 = 3 - 1
[2,1,3,4] => 011 => 100 => 1 = 2 - 1
[2,1,4,3] => 010 => 101 => 1 = 2 - 1
[2,3,1,4] => 001 => 110 => 2 = 3 - 1
[2,3,4,1] => 000 => 111 => 3 = 4 - 1
[2,4,1,3] => 000 => 111 => 3 = 4 - 1
[2,4,3,1] => 000 => 111 => 3 = 4 - 1
[3,1,2,4] => 001 => 110 => 2 = 3 - 1
[3,1,4,2] => 000 => 111 => 3 = 4 - 1
[3,2,1,4] => 001 => 110 => 2 = 3 - 1
[3,2,4,1] => 000 => 111 => 3 = 4 - 1
[3,4,1,2] => 000 => 111 => 3 = 4 - 1
[3,4,2,1] => 000 => 111 => 3 = 4 - 1
[4,1,2,3] => 000 => 111 => 3 = 4 - 1
[4,1,3,2] => 000 => 111 => 3 = 4 - 1
[4,2,1,3] => 000 => 111 => 3 = 4 - 1
[4,2,3,1] => 000 => 111 => 3 = 4 - 1
[4,3,1,2] => 000 => 111 => 3 = 4 - 1
[4,3,2,1] => 000 => 111 => 3 = 4 - 1
[1,2,3,4,5] => 1111 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 1110 => 0001 => 1 = 2 - 1
[1,2,4,3,5] => 1101 => 0010 => 1 = 2 - 1
[1,2,4,5,3] => 1100 => 0011 => 2 = 3 - 1
[1,2,5,3,4] => 1100 => 0011 => 2 = 3 - 1
[1,2,5,4,3] => 1100 => 0011 => 2 = 3 - 1
[1,3,2,4,5] => 1011 => 0100 => 1 = 2 - 1
[1,3,2,5,4] => 1010 => 0101 => 1 = 2 - 1
[1,3,4,2,5] => 1001 => 0110 => 2 = 3 - 1
[1,3,4,5,2] => 1000 => 0111 => 3 = 4 - 1
[1,3,5,2,4] => 1000 => 0111 => 3 = 4 - 1
[1,3,5,4,2] => 1000 => 0111 => 3 = 4 - 1
[1,4,2,3,5] => 1001 => 0110 => 2 = 3 - 1
[1,4,2,5,3] => 1000 => 0111 => 3 = 4 - 1
[1,4,3,2,5] => 1001 => 0110 => 2 = 3 - 1
[1,4,3,5,2] => 1000 => 0111 => 3 = 4 - 1
[1,4,5,2,3] => 1000 => 0111 => 3 = 4 - 1
[1,4,5,3,2] => 1000 => 0111 => 3 = 4 - 1
[8,6,5,4,7,3,2,1] => ? => ? => ? = 8 - 1
[8,5,6,4,7,3,2,1] => ? => ? => ? = 8 - 1
[8,6,4,5,7,3,2,1] => ? => ? => ? = 8 - 1
[8,4,5,6,7,3,2,1] => ? => ? => ? = 8 - 1
[8,7,6,4,3,5,2,1] => ? => ? => ? = 8 - 1
[7,8,3,4,5,6,2,1] => ? => ? => ? = 8 - 1
[7,4,5,6,3,8,2,1] => ? => ? => ? = 8 - 1
[6,4,5,7,3,8,2,1] => ? => ? => ? = 8 - 1
[7,5,3,4,6,8,2,1] => ? => ? => ? = 8 - 1
[7,3,4,5,6,8,2,1] => ? => ? => ? = 8 - 1
[6,5,3,4,7,8,2,1] => ? => ? => ? = 8 - 1
[6,3,4,5,7,8,2,1] => ? => ? => ? = 8 - 1
[8,4,5,6,7,2,3,1] => ? => ? => ? = 8 - 1
[8,7,6,5,3,2,4,1] => ? => ? => ? = 8 - 1
[8,7,6,4,3,2,5,1] => ? => ? => ? = 8 - 1
[8,7,6,3,4,2,5,1] => ? => ? => ? = 8 - 1
[7,8,6,3,2,4,5,1] => ? => ? => ? = 8 - 1
[8,7,5,4,3,2,6,1] => ? => ? => ? = 8 - 1
[8,7,5,3,4,2,6,1] => ? => ? => ? = 8 - 1
[8,7,5,3,2,4,6,1] => ? => ? => ? = 8 - 1
[7,8,3,4,2,5,6,1] => ? => ? => ? = 8 - 1
[7,8,4,2,3,5,6,1] => ? => ? => ? = 8 - 1
[8,6,5,3,4,2,7,1] => ? => ? => ? = 8 - 1
[8,4,3,5,6,2,7,1] => ? => ? => ? = 8 - 1
[8,6,2,3,4,5,7,1] => ? => ? => ? = 8 - 1
[8,3,4,5,2,6,7,1] => ? => ? => ? = 8 - 1
[8,5,4,2,3,6,7,1] => ? => ? => ? = 8 - 1
[8,4,2,3,5,6,7,1] => ? => ? => ? = 8 - 1
[7,5,6,4,3,2,8,1] => ? => ? => ? = 8 - 1
[7,4,5,6,3,2,8,1] => ? => ? => ? = 8 - 1
[6,4,5,7,3,2,8,1] => ? => ? => ? = 8 - 1
[7,6,5,3,4,2,8,1] => ? => ? => ? = 8 - 1
[7,4,5,3,6,2,8,1] => ? => ? => ? = 8 - 1
[7,5,3,4,6,2,8,1] => ? => ? => ? = 8 - 1
[4,5,6,3,7,2,8,1] => ? => ? => ? = 8 - 1
[4,5,3,6,7,2,8,1] => ? => ? => ? = 8 - 1
[5,6,4,7,2,3,8,1] => ? => ? => ? = 8 - 1
[7,6,3,2,4,5,8,1] => ? => ? => ? = 8 - 1
[7,5,4,2,3,6,8,1] => ? => ? => ? = 8 - 1
[7,5,2,3,4,6,8,1] => ? => ? => ? = 8 - 1
[7,3,2,4,5,6,8,1] => ? => ? => ? = 8 - 1
[4,5,6,3,2,7,8,1] => ? => ? => ? = 8 - 1
[6,5,3,4,2,7,8,1] => ? => ? => ? = 8 - 1
[6,5,7,8,4,3,1,2] => ? => ? => ? = 8 - 1
[8,5,4,6,7,3,1,2] => ? => ? => ? = 8 - 1
[5,6,4,7,8,3,1,2] => ? => ? => ? = 8 - 1
[8,7,3,4,5,6,1,2] => ? => ? => ? = 8 - 1
[8,5,3,4,6,7,1,2] => ? => ? => ? = 8 - 1
[8,4,5,6,7,2,1,3] => ? => ? => ? = 8 - 1
Description
The length of the longest run of ones in a binary word.
Mp00159: Permutations Demazure product with inversePermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00117: Graphs Ore closureGraphs
St001330: Graphs ⟶ ℤResult quality: 69% values known / values provided: 69%distinct values known / distinct values provided: 80%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([],2)
=> ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [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)
=> 4
[2,4,1,3] => [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)
=> 4
[2,4,3,1] => [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)
=> 4
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => [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)
=> 4
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [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)
=> 4
[3,4,1,2] => [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)
=> 4
[3,4,2,1] => [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)
=> 4
[4,1,2,3] => [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)
=> 4
[4,1,3,2] => [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)
=> 4
[4,2,1,3] => [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)
=> 4
[4,2,3,1] => [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)
=> 4
[4,3,1,2] => [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)
=> 4
[4,3,2,1] => [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)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[5,1,2,4,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,4,3,5,6,2] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,5,2,3,6,4] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,5,2,4,6,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,6,2,3,4,5] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,6,2,3,5,4] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,3,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,3,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,3,4,6,5,1] => [6,2,3,5,4,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,3,5,4,6,1] => [6,2,4,3,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,4,1,6,3,5] => [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),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 6
[2,4,3,5,6,1] => [6,3,2,4,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,6,1,3,4,5] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,1,4,5,2,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,1,4,5,6,2] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,1,4,6,5,2] => [6,2,3,5,4,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,1,5,4,6,2] => [6,2,4,3,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,2,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,2,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,2,4,6,5,1] => [6,2,3,5,4,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,2,5,4,6,1] => [6,2,4,3,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,4,1,5,6,2] => [6,3,2,4,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[3,4,2,5,6,1] => [6,3,2,4,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,2,5,3,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,2,5,6,3] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,2,6,3,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,2,6,5,3] => [6,2,3,5,4,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,3,5,2,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,3,5,6,2] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,3,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,3,6,5,2] => [6,2,3,5,4,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,5,2,6,3] => [6,2,4,3,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,1,5,3,6,2] => [6,2,4,3,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,2,1,5,6,3] => [6,3,2,4,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[4,2,3,5,6,1] => [6,3,2,4,5,1] => ([(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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001120The length of a longest path in a graph. St001268The size of the largest ordinal summand in the poset. St000171The degree of the graph. St001645The pebbling number of a connected graph. St000209Maximum difference of elements in cycles. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement 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. St001090The number of pop-stack-sorts needed to sort a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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).