Your data matches 213 different statistics following compositions of up to 3 maps.
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St000998: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2
[1,0,1,0]
=> 3
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 5
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Mp00201: Dyck paths RingelPermutations
St001004: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 2
[1,0,1,0]
=> [3,1,2] => 3
[1,1,0,0]
=> [2,3,1] => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => 4
[1,1,0,0,1,0]
=> [2,4,1,3] => 4
[1,1,0,1,0,0]
=> [4,3,1,2] => 3
[1,1,1,0,0,0]
=> [2,3,4,1] => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 4
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5
Description
The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Mp00201: Dyck paths RingelPermutations
St001005: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 2
[1,0,1,0]
=> [3,1,2] => 3
[1,1,0,0]
=> [2,3,1] => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => 4
[1,1,0,0,1,0]
=> [2,4,1,3] => 4
[1,1,0,1,0,0]
=> [4,3,1,2] => 3
[1,1,1,0,0,0]
=> [2,3,4,1] => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 4
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 4
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 5
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 5
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
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
St001012: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 2
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 3
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 4
[1,1,1,0,0,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 5
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 5
[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]
=> 3
[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]
=> 4
[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]
=> 5
[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]
=> 4
[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]
=> 5
[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]
=> 5
[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]
=> 6
[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]
=> 4
[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]
=> 4
[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]
=> 5
[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]
=> 6
[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]
=> 6
[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]
=> 6
[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]
=> 6
[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]
=> 4
[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]
=> 5
[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]
=> 6
[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]
=> 5
[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]
=> 6
[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]
=> 4
[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]
=> 5
[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]
=> 4
[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]
=> 4
[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]
=> 5
[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]
=> 6
[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]
=> 5
[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]
=> 5
[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]
=> 6
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
Description
The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St001318
Mp00242: Dyck paths Hessenberg posetPosets
Mp00198: Posets incomparability graphGraphs
Mp00157: Graphs connected complementGraphs
St001318: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> 2 = 3 - 1
[1,1,0,0]
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> ([],3)
=> ([(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]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> ([],4)
=> ([(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)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],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)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([(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
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St001246: Permutations ⟶ ℤResult quality: 80% values known / values provided: 98%distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1] => [1] => ? = 2 - 2
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 1 = 3 - 2
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => [1,2,3] => 1 = 3 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [2,1,3] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 2 = 4 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,3,2] => 2 = 4 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [2,3,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => [2,1,3,4] => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,2,3,1] => [2,3,1,4] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => [1,2,3,4] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [1,3,2,4] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,3,2,1] => [3,2,1,4] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [1,4,2,3] => 3 = 5 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => [3,1,2,4,5] => 2 = 4 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,2,4,1] => [3,1,4,2,5] => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,3,2,5,1] => [3,2,1,4,5] => 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,3,2,4,1] => [3,2,4,1,5] => 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,2,4,3,1] => [2,4,3,1,5] => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,5,1] => [2,3,1,4,5] => 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,2,3,4,1] => [2,3,4,1,5] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => [2,1,3,4,5] => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,4,1] => [2,1,4,3,5] => 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,2,3,1] => [3,4,2,1,5] => 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => [1,2,3,4,5] => 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => [1,2,4,3,5] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,4,3,1] => [1,4,3,2,5] => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [4,3,2,1,5] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,4,1,5,2] => [5,1,2,4,3] => 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,5,1,4,2] => [5,1,4,2,3] => 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,3,1,5,2] => [5,2,1,4,3] => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,3,1,4,2] => [5,2,4,1,3] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [5,4,3,1,2] => 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,5,1,3] => [2,5,1,3,4] => 4 = 6 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,2,4,1,3] => [2,5,3,1,4] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,1,4] => [2,1,5,3,4] => 4 = 6 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => [2,1,3,5,4] => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,2,3,1,5] => [2,3,1,5,4] => 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => [1,2,3,5,4] => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,3,1,5] => [1,3,2,5,4] => 3 = 5 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,5,2,3] => [4,5,1,3,2] => 4 = 6 - 2
Description
The maximal difference between two consecutive entries of a permutation. This is given, for a permutation $\pi$ of length $n$, by $$\max\{ | \pi(i) - \pi(i+1) | : 1 \leq i < n \}.$$
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00160: Permutations graph of inversionsGraphs
St000777: Graphs ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {4,4} - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {4,4} - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {4,4,5,5,5} - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {4,4,5,5,5} - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {4,4,5,5,5} - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {4,4,5,5,5} - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {4,4,5,5,5} - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [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)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,6,6} - 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St001232: Dyck paths ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4} - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {4,4} - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5} - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,5,5,5} - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5} - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {4,4,5,5,5} - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5} - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5,5,5,5,6} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Mp00199: Dyck paths prime Dyck pathDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
Mp00232: Dyck paths parallelogram posetPosets
St000907: Posets ⟶ ℤResult quality: 62% values known / values provided: 62%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {3,4}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {3,4}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
Description
The number of maximal antichains of minimal length in a poset.
The following 203 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001863The number of weak excedances of a signed permutation. St000259The diameter of a connected graph. St000717The number of ordinal summands of a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000141The maximum drop size of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000054The first entry of the permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000381The largest part of an integer composition. St000260The radius of a connected graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000653The last descent of a permutation. St000808The number of up steps of the associated bargraph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000765The number of weak records in an integer composition. St000942The number of critical left to right maxima of the parking functions. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001777The number of weak descents in an integer composition. St001935The number of ascents in a parking function. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St000834The number of right outer peaks of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000521The number of distinct subtrees of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St000670The reversal length of a permutation. St001861The number of Bruhat lower covers of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000568The hook number of a binary tree. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000050The depth or height of a binary tree. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St000035The number of left outer peaks of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000264The girth of a graph, which is not a tree. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000451The length of the longest pattern of the form k 1 2. St000028The number of stack-sorts needed to sort a permutation. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001394The genus of a permutation. St000094The depth of an ordered tree. St000422The energy of a graph, if it is integral. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St001114The number of odd descents of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001928The number of non-overlapping descents in a permutation. St001469The holeyness of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000702The number of weak deficiencies of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001267The length of the Lyndon factorization of the binary word. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001391The disjunction number of a graph. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001927Sparre Andersen's number of positives of a signed permutation. St000021The number of descents of a permutation. St000080The rank of the poset. St000089The absolute variation of a composition. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000239The number of small weak excedances. St000273The domination number of a graph. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000489The number of cycles of a permutation of length at most 3. St000495The number of inversions of distance at most 2 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000906The length of the shortest maximal chain in a poset. St000916The packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001874Lusztig's a-function for the symmetric group. St001955The number of natural descents for set-valued two row standard Young tableaux. St000023The number of inner peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001388The number of non-attacking neighbors of a permutation. St001470The cyclic holeyness of a permutation. St001520The number of strict 3-descents. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001839The number of excedances of a set partition. St000806The semiperimeter of the associated bargraph. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000527The width of the poset. St000528The height of a poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000488The number of cycles of a permutation of length at most 2. St000632The jump number of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001820The size of the image of the pop stack sorting operator. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001569The maximal modular displacement of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. 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. St001510The number of self-evacuating linear extensions of a finite poset. St001782The order of rowmotion on the set of order ideals of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000307The number of rowmotion orbits of a poset. St001555The order of a signed permutation. St001644The dimension of a graph. St001812The biclique partition number of a graph. St001638The book thickness of a graph.