Your data matches 262 different statistics following compositions of up to 3 maps.
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Mp00114: Permutations connectivity setBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000259: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => => [1] => ([],1)
=> 0
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,3,5] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,2,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,2,1,4,5] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,4,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,2,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,3,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,1,3,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,3,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,1,2,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,2,1,5] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,3,4,5,6] => 11111 => [1,1,1,1,1,1] => ([(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)
=> 1
[1,2,3,5,4,6] => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,4,3,5,6] => 11011 => [1,1,2,1,1] => ([(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)
=> 2
[1,2,4,5,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,3,4,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,4,3,6] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2,4,5,6] => 10111 => [1,2,1,1,1] => ([(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)
=> 2
[1,3,2,5,4,6] => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,4,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,4,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,5,2,4,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,5,4,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,3,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,2,5,3,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,2,5,6] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,5,2,6] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Mp00061: Permutations to increasing treeBinary trees
Mp00013: Binary trees to posetPosets
St000846: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> ([],1)
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,3,5,2,4,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,4,2,3,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[] => .
=> ?
=> ? = 0
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000147
Mp00252: Permutations restrictionPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00108: Permutations cycle typeInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => []
=> 0
[1,2] => [1] => [1] => [1]
=> 1
[1,2,3] => [1,2] => [1,2] => [1,1]
=> 1
[2,1,3] => [2,1] => [2,1] => [2]
=> 2
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1,4] => [2,3,1] => [3,2,1] => [2,1]
=> 2
[3,1,2,4] => [3,1,2] => [3,2,1] => [2,1]
=> 2
[3,2,1,4] => [3,2,1] => [3,2,1] => [2,1]
=> 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2
[2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4,5] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 2
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => [2,2]
=> 2
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => [2,2]
=> 2
[3,1,2,4,5] => [3,1,2,4] => [3,2,1,4] => [2,1,1]
=> 2
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => [2,1,1]
=> 2
[3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => [2,1,1]
=> 2
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => [2,2]
=> 2
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => [2,2]
=> 2
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => [2,1,1]
=> 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => [2,1,1]
=> 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => [2,2]
=> 2
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => [2,2]
=> 2
[4,3,1,2,5] => [4,3,1,2] => [4,3,2,1] => [2,2]
=> 2
[4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => [2,2]
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 2
[1,2,4,3,5,6] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 2
[1,2,4,5,3,6] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 2
[1,2,5,3,4,6] => [1,2,5,3,4] => [1,2,5,4,3] => [2,1,1,1]
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3] => [1,2,5,4,3] => [2,1,1,1]
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5,6] => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,3,4,5,2,6] => [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4] => [1,4,5,2,3] => [2,2,1]
=> 2
[1,3,5,4,2,6] => [1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> 2
[1,4,2,3,5,6] => [1,4,2,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,4,2,5,3,6] => [1,4,2,5,3] => [1,5,3,4,2] => [2,1,1,1]
=> 2
[1,4,3,2,5,6] => [1,4,3,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,4,3,5,2,6] => [1,4,3,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 2
[] => ? => ? => ?
=> ? = 0
Description
The largest part of an integer partition.
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000793: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 0
[1,2] => [2] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[3,1,4,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[3,2,4,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[3,4,2,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[4,1,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[4,2,1,3,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[4,2,3,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[4,3,1,2,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[4,3,2,1,5] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 1
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 2
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
[1,2,4,5,3,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 2
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 2
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
[1,3,4,5,2,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 2
[1,3,5,2,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2
[1,4,2,5,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 2
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2
[1,4,3,5,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 2
[] => [] => ?
=> ?
=> ? = 0
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$. Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row. Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$. This statistic is the length of the longest partition on the diagonal of the diagram.
Mp00061: Permutations to increasing treeBinary trees
Mp00013: Binary trees to posetPosets
Mp00125: Posets dual posetPosets
St000845: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> ([],1)
=> ([],1)
=> 0
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 2
[1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> 2
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,3,5,2,4,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> 2
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,4,2,3,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 2
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> 2
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[] => .
=> ?
=> ?
=> ? = 0
Description
The maximal number of elements covered by an element in a poset.
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,2] => [2]
=> [1,0,1,0]
=> [1,1] => 2 = 1 + 1
[1,2,3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 2 = 1 + 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 3 = 2 + 1
[1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 2 = 1 + 1
[1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 3 = 2 + 1
[1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 2 = 1 + 1
[1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[2,4,1,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[3,1,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[3,1,4,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[3,4,1,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[4,1,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 3 = 2 + 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[4,3,1,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 3 = 2 + 1
[1,2,3,4,5,6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 2 = 1 + 1
[1,2,3,5,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,2,4,3,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,2,4,5,3,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,2,5,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,2,5,4,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 3 = 2 + 1
[1,3,2,4,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,3,2,5,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,3,4,2,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,3,4,5,2,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,3,5,2,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,3,5,4,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 3 = 2 + 1
[1,4,2,3,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 3 = 2 + 1
[1,4,2,5,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,4,3,2,5,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 3 = 2 + 1
[1,4,3,5,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 3 = 2 + 1
[] => []
=> []
=> [] => ? = 0 + 1
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Mp00252: Permutations restrictionPermutations
Mp00159: Permutations Demazure product with inversePermutations
St000058: Permutations ⟶ ℤResult quality: 67% values known / values provided: 100%distinct values known / distinct values provided: 67%
Values
[1] => [] => [] => ? = 0
[1,2] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => [3,2,1] => 2
[3,1,2,4] => [3,1,2] => [3,2,1] => 2
[3,2,1,4] => [3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4,5] => [2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => 2
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => 2
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => 2
[3,1,2,4,5] => [3,1,2,4] => [3,2,1,4] => 2
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => 2
[3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => 2
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => 2
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => 2
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => 2
[4,2,1,3,5] => [4,2,1,3] => [4,3,2,1] => 2
[4,2,3,1,5] => [4,2,3,1] => [4,3,2,1] => 2
[4,3,1,2,5] => [4,3,1,2] => [4,3,2,1] => 2
[4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5,6] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4,6] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5,6] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3,6] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,2,5,3,4,6] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,2,5,4,3,6] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5,6] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4,6] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5,6] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2,6] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4,6] => [1,3,5,2,4] => [1,4,5,2,3] => 2
[1,3,5,4,2,6] => [1,3,5,4,2] => [1,5,4,3,2] => 2
[1,4,2,3,5,6] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,4,2,5,3,6] => [1,4,2,5,3] => [1,5,3,4,2] => 2
[1,4,3,2,5,6] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2,6] => [1,4,3,5,2] => [1,5,3,4,2] => 2
[1,4,5,2,3,6] => [1,4,5,2,3] => [1,5,4,3,2] => 2
[] => ? => ? => ? = 0
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Mp00252: Permutations restrictionPermutations
Mp00065: Permutations permutation posetPosets
St000298: Posets ⟶ ℤResult quality: 67% values known / values provided: 100%distinct values known / distinct values provided: 67%
Values
[1] => [] => ([],0)
=> ? = 0
[1,2] => [1] => ([],1)
=> 1
[1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[2,1,3] => [2,1] => ([],2)
=> 2
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[3,2,1,4] => [3,2,1] => ([],3)
=> 2
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,2,3,5] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4,5] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1,5] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[3,1,2,4,5] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1,5] => [3,4,2,1] => ([(2,3)],4)
=> 2
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,1,3,2,5] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 2
[4,2,1,3,5] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1,5] => [4,2,3,1] => ([(2,3)],4)
=> 2
[4,3,1,2,5] => [4,3,1,2] => ([(2,3)],4)
=> 2
[4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,4,5,3,6] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,3,4,6] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,3,5,4,2,6] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2,6] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[] => ? => ?
=> ? = 0
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Mp00159: Permutations Demazure product with inversePermutations
Mp00108: Permutations cycle typeInteger partitions
St000668: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 100%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1]
=> ? = 0
[1,2] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2
[2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> 2
[2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[2,4,1,3,5] => [3,4,1,2,5] => [2,2,1]
=> 2
[2,4,3,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[3,1,2,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> 2
[3,1,4,2,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> 2
[3,2,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[3,4,2,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[4,1,2,3,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[4,1,3,2,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[4,2,1,3,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[4,2,3,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[4,3,1,2,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [2,2,1]
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,1,1,1,1]
=> 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,1,1,1,1]
=> 2
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [2,1,1,1,1]
=> 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [2,1,1,1,1]
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [2,1,1,1,1]
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [2,1,1,1,1]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,1,1]
=> 2
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [2,1,1,1,1]
=> 2
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [2,1,1,1,1]
=> 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [2,2,1,1]
=> 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [2,2,1,1]
=> 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [2,1,1,1,1]
=> 2
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [2,1,1,1,1]
=> 2
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [2,1,1,1,1]
=> 2
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [2,1,1,1,1]
=> 2
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [2,2,1,1]
=> 2
[] => [] => []
=> ? = 0
Description
The least common multiple of the parts of the partition.
Mp00159: Permutations Demazure product with inversePermutations
Mp00151: Permutations to cycle typeSet partitions
St001062: Set partitions ⟶ ℤResult quality: 67% values known / values provided: 100%distinct values known / distinct values provided: 67%
Values
[1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => {{1},{2}}
=> 1
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2
[2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 2
[2,4,1,3,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 2
[2,4,3,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2
[3,1,4,2,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 2
[3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2
[3,2,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[3,4,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[4,1,2,3,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 2
[4,1,3,2,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 2
[4,2,1,3,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[4,2,3,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[4,3,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}}
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}}
=> 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => {{1},{2},{3,4},{5},{6}}
=> 2
[1,2,4,5,3,6] => [1,2,5,4,3,6] => {{1},{2},{3,5},{4},{6}}
=> 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => {{1},{2},{3,5},{4},{6}}
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => {{1},{2},{3,5},{4},{6}}
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => {{1},{2,3},{4},{5},{6}}
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => {{1},{2,3},{4,5},{6}}
=> 2
[1,3,4,2,5,6] => [1,4,3,2,5,6] => {{1},{2,4},{3},{5},{6}}
=> 2
[1,3,4,5,2,6] => [1,5,3,4,2,6] => {{1},{2,5},{3},{4},{6}}
=> 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => {{1},{2,4},{3,5},{6}}
=> 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => {{1},{2,5},{3,4},{6}}
=> 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => {{1},{2,4},{3},{5},{6}}
=> 2
[1,4,2,5,3,6] => [1,5,3,4,2,6] => {{1},{2,5},{3},{4},{6}}
=> 2
[1,4,3,2,5,6] => [1,4,3,2,5,6] => {{1},{2,4},{3},{5},{6}}
=> 2
[1,4,3,5,2,6] => [1,5,3,4,2,6] => {{1},{2,5},{3},{4},{6}}
=> 2
[1,4,5,2,3,6] => [1,5,4,3,2,6] => {{1},{2,5},{3,4},{6}}
=> 2
[] => [] => {}
=> ? = 0
Description
The maximal size of a block of a set partition.
The following 252 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001128The exponens consonantiae of a partition. St000253The crossing number of a set partition. St000392The length of the longest run of ones in a binary word. St000659The number of rises of length at least 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000402Half the size of the symmetry class of a permutation. St000444The length of the maximal rise of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000630The length of the shortest palindromic decomposition of a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St000983The length of the longest alternating subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001093The detour number of a graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St001674The number of vertices of the largest induced star graph in the graph. St000254The nesting number of a set partition. St000260The radius of a connected graph. St000292The number of ascents of a binary word. St000297The number of leading ones in a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000535The rank-width of a graph. St000691The number of changes of a binary word. St000730The maximal arc length of a set partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000919The number of maximal left branches of a binary tree. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001271The competition number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000485The length of the longest cycle of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000264The girth of a graph, which is not a tree. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001657The number of twos in an integer partition. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001471The magnitude of a Dyck path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000517The Kreweras number of an integer partition. St000640The rank of the largest boolean interval in a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000914The sum of the values of the Möbius function of a poset. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001593This is the number of standard Young tableaux of the given shifted shape. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000455The second largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000921The number of internal inversions of a binary word. St000929The constant term of the character polynomial of an integer partition. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001281The normalized isoperimetric number of a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001570The minimal number of edges to add to make a graph Hamiltonian. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000225Difference between largest and smallest parts in a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St000944The 3-degree of an integer partition. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001890The maximum magnitude of the Möbius function of a poset. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001793The difference between the clique number and the chromatic number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001307The number of induced stars on four vertices in a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001060The distinguishing index of a graph. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000647The number of big descents of a permutation. St000940The number of characters of the symmetric group whose value on the partition is zero. St000451The length of the longest pattern of the form k 1 2. St000486The number of cycles of length at least 3 of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000996The number of exclusive left-to-right maxima of a permutation. St000353The number of inner valleys of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000472The sum of the ascent bottoms of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000454The largest eigenvalue of a graph if it is integral. St001555The order of a signed permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001488The number of corners of a skew partition. St000893The number of distinct diagonal sums of an alternating sign matrix. 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 $eA$. St001569The maximal modular displacement of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001624The breadth of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation.