Your data matches 51 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000308
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00064: Permutations reversePermutations
St000308: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [2,4,3,5,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,4,3,5,2] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => 4
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St000013
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000094
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00015: Binary trees to ordered tree: right child = right brotherOrdered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [[]]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [[],[]]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [[[]]]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [[[],[]]]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[],[[],[],[]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[],[[],[[]]]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[],[[[]],[]]]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[],[[[]],[]]]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[],[[[[]]]]]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[[]],[[],[]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[[]],[[[]]]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[[],[]],[[]]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[],[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[[],[],[],[]]]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [[[],[],[[]]]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[[],[[]]],[]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [[[],[[]],[]]]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [[[],[[]],[]]]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [[[],[[[]]]]]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [[[],[],[]],[[[]]]]
=> ? = 3 + 1
Description
The depth of an ordered tree.
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00066: Permutations inversePermutations
Mp00241: Permutations invert Laguerre heapPermutations
St000451: Permutations ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [3,1,2] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,2,3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [3,4,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 3
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => [1,4,5,2,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,5,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [2,5,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [4,2,3,1,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [4,5,2,3,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [3,5,2,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [5,2,3,1,4] => 4
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 5
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,5,6] => [4,1,2,3,6,7,5] => [2,3,4,1,7,5,6] => ? = 3
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000028
Mp00028: Dyck paths reverseDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00241: Permutations invert Laguerre heapPermutations
St000028: Permutations ⟶ ℤResult quality: 95% values known / values provided: 95%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,4,1,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,4,1,3,5] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,4,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [5,2,4,1,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,5,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7] => [1,3,4,2,5,6,7] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => [1,3,4,5,2,6,7] => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => ? = 4 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => [1,3,4,5,6,2,7] => ? = 5 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,1,2,5,6,7,4] => [2,3,1,7,4,5,6] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 4 - 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => ? = 5 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [1,3,4,5,6,7,2] => ? = 6 - 1
Description
The number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
St000062: Permutations ⟶ ℤResult quality: 86% values known / values provided: 92%distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [7,4,5,6,3,2,1] => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [7,6,3,4,5,2,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ? = 5
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [7,6,5,2,3,4,1] => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [7,6,2,3,4,5,1] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => ? = 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? = 6
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [5,6,7,3,2,1,4] => ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [[[[[.,.],.],.],.],[.,[.,.]]]
=> [7,6,1,2,3,4,5] => ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => ? = 7
Description
The length of the longest increasing subsequence of the permutation.
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00015: Binary trees to ordered tree: right child = right brotherOrdered trees
St000166: Ordered trees ⟶ ℤResult quality: 86% values known / values provided: 92%distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [.,.]
=> [[]]
=> 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [[],[]]
=> 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [[[]]]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [[[],[]]]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 3
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[],[[],[],[]]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[],[[],[[]]]]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[],[[[]],[]]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[],[[[]],[]]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[],[[[[]]]]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[[]],[],[[]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[]],[[]],[]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[[]],[[],[]]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[[]],[[[]]]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[[],[]],[[]]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[],[]],[]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[[],[],[],[]]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [[[],[],[[]]]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[[],[[]]],[]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [[[],[[]],[]]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [[[],[[]],[]]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [[[],[[[]]]]]
=> 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [[],[],[],[],[[[]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[],[],[[[]]],[]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [[],[],[],[[[[]]]]]
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[],[],[[[]]],[],[]]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [[],[],[[[[]]]],[]]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]]
=> [[],[],[[[[[]]]]]]
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [[],[[[]]],[],[],[]]
=> ? = 3
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [[],[[[[]]]],[],[]]
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [[],[[[[[]]]]],[]]
=> ? = 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [[],[[[[[[]]]]]]]
=> ? = 6
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [[[],[],[]],[[[]]]]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [[[[]]],[],[],[],[]]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[[]]]],[],[],[]]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [[[[[.,.],.],.],.],[.,[.,.]]]
=> [[[[[[]]]]],[],[]]
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [[[[[[[]]]]]],[]]
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [[[[[[[[]]]]]]]]
=> ? = 7
Description
The depth minus 1 of an ordered tree. The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St001644: Graphs ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [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)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(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)
=> ([(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)
=> ? = 4 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => ([(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)
=> ([(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)
=> ? = 4 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => ([(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)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
Description
The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St001330: Graphs ⟶ ℤResult quality: 61% values known / values provided: 61%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,1,0,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [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
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(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
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
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 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St000381The largest part of an integer composition. St000877The depth of the binary word interpreted as a path. St000914The sum of the values of the Möbius function of a poset. St000141The maximum drop size of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001717The largest size of an interval in a poset. St001890The maximum magnitude of the Möbius function of a poset. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000983The length of the longest alternating subword. St000662The staircase size of the code of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000097The order of the largest clique of the graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000317The cycle descent number of a permutation. St000628The balance of a binary word. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000080The rank of the poset. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001590The crossing number of a perfect matching. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001624The breadth of a lattice.