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Your data matches 151 different statistics following compositions of up to 3 maps.
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Matching statistic: St001554
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,[.,.]],.],.],.]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,[[[.,.],.],.]]],.]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,[[.,.],.]],.]],.]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],.]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[[[.,[.,.]],.],.]],.]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[.,[[[[.,.],.],.],.]],.]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[[.,[.,[[.,.],.]]],.],.]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],.]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[[.,[[.,[.,.]],.]],.],.]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[[.,[[[.,.],.],.]],.],.]
=> 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,[[.,.],.]],.],.],.]
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> 7
Description
The number of distinct nonempty subtrees of a binary tree.
Matching statistic: St000998
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[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]
=> 6 = 5 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[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]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[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]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 8 = 7 + 1
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.
Matching statistic: St001004
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => [2,1] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 3 = 4 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,4,1,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,1,3,5] => 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,4,1,3] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [2,1,5,3,4] => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,4,1,2,5] => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,1,2] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,5,2,4] => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,1,2,4,5] => 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [1,4,2,3,5] => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => 4 = 5 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,3,4,5,1,6] => 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [2,3,4,6,1,5] => 6 = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [2,3,4,1,5,6] => 6 = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [2,3,5,6,1,4] => 6 = 7 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [2,3,5,1,4,6] => 6 = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [2,3,1,6,4,5] => 6 = 7 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [2,3,1,4,5,6] => 6 = 7 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [2,4,5,6,1,3] => 6 = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [2,4,5,1,3,6] => 6 = 7 - 1
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]].
Matching statistic: St000203
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [.,[.,.]]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [[[[.,.],.],.],[.,[.,.]]]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [[[[.,.],.],[.,.]],[.,.]]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [[[[.,.],.],.],[.,[.,.]]]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [[[[.,.],.],.],[.,[.,.]]]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [[[.,.],[.,.]],[.,[.,.]]]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => [[[.,.],[.,.]],[.,[.,.]]]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [[[[[.,.],.],.],.],[.,[.,.]]]
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [[[[[.,.],.],.],.],[.,[.,.]]]
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [[[[[.,.],.],.],.],[.,[.,.]]]
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [[[[[.,.],.],.],.],[.,[.,.]]]
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 7
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000459
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> {{1,2}}
=> [2]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> [2,2,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> [3,2]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> [4,1]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> [3,2]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> [4,1]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> [5]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> {{1,6},{2},{3},{4,5}}
=> [2,2,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> {{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,4,5,6},{2},{3}}
=> [4,1,1]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,3,5,6},{2},{4}}
=> [4,1,1]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> {{1,3,6},{2},{4,5}}
=> [3,2,1]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,3,4,6},{2},{5}}
=> [4,1,1]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,3,4,5,6},{2}}
=> [5,1]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,2,5,6},{3},{4}}
=> [4,1,1]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> {{1,2,6},{3},{4,5}}
=> [3,2,1]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> {{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> {{1,2,4,5,6},{3}}
=> [5,1]
=> 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,5,6},{4}}
=> [5,1]
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,6},{4,5}}
=> [4,2]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,6},{5}}
=> [5,1]
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> [6]
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,6,7},{2},{3},{4},{5}}
=> [3,1,1,1,1]
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> {{1,5,7},{2},{3},{4},{6}}
=> [3,1,1,1,1]
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,5,6,7},{2},{3},{4}}
=> [4,1,1,1]
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,4,7},{2},{3},{5},{6}}
=> [3,1,1,1,1]
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,4,6,7},{2},{3},{5}}
=> [4,1,1,1]
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,4,5,7},{2},{3},{6}}
=> [4,1,1,1]
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,4,5,6,7},{2},{3}}
=> [5,1,1]
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,3,7},{2},{4},{5},{6}}
=> [3,1,1,1,1]
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,3,6,7},{2},{4},{5}}
=> [4,1,1,1]
=> 7
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000528
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(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]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(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]
=> [[.,[.,[[.,.],.]]],.]
=> ([(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]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(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]
=> [[.,[[[.,.],.],.]],.]
=> ([(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]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(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]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(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]
=> [[[[[.,.],.],.],.],.]
=> ([(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]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(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]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(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]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(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]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(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]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[[[.,[.,.]],.],.]],.]
=> ([(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]
=> [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(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]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(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]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(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]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(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]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St000725
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 5
[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]
=> [1,3,4,5,2] => 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 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]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,4,2] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 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]
=> [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,6,3,2] => 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,3,2] => 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,4,2] => 7
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000870
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> {{1,2}}
=> [2]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> [2,2,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> [3,2]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> [4,1]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> [3,2]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> [4,1]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> [5]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> {{1,6},{2},{3},{4,5}}
=> [2,2,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> {{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,4,5,6},{2},{3}}
=> [4,1,1]
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,3,5,6},{2},{4}}
=> [4,1,1]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> {{1,3,6},{2},{4,5}}
=> [3,2,1]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,3,4,6},{2},{5}}
=> [4,1,1]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,3,4,5,6},{2}}
=> [5,1]
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,2,5,6},{3},{4}}
=> [4,1,1]
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> {{1,2,6},{3},{4,5}}
=> [3,2,1]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> {{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> {{1,2,4,5,6},{3}}
=> [5,1]
=> 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,5,6},{4}}
=> [5,1]
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,6},{4,5}}
=> [4,2]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,6},{5}}
=> [5,1]
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> [6]
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,6,7},{2},{3},{4},{5}}
=> [3,1,1,1,1]
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> {{1,5,7},{2},{3},{4},{6}}
=> [3,1,1,1,1]
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,5,6,7},{2},{3},{4}}
=> [4,1,1,1]
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,4,7},{2},{3},{5},{6}}
=> [3,1,1,1,1]
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,4,6,7},{2},{3},{5}}
=> [4,1,1,1]
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,4,5,7},{2},{3},{6}}
=> [4,1,1,1]
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,4,5,6,7},{2},{3}}
=> [5,1,1]
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,3,7},{2},{4},{5},{6}}
=> [3,1,1,1,1]
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,3,6,7},{2},{4},{5}}
=> [4,1,1,1]
=> 7
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St000907
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
[1,1,0,1,0,1,0,0]
=> [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
[1,1,1,0,0,1,0,0]
=> [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
[1,0,1,1,0,1,0,1,0,0]
=> [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
[1,0,1,1,1,0,0,1,0,0]
=> [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
[1,1,0,1,0,1,0,1,0,0]
=> [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
[1,1,0,1,1,0,0,1,0,0]
=> [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
[1,1,1,0,1,0,1,0,0,0]
=> [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
[1,1,1,1,0,0,1,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000912
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(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]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(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]
=> [[.,[.,[[.,.],.]]],.]
=> ([(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]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(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]
=> [[.,[[[.,.],.],.]],.]
=> ([(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]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> ([(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]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,[.,.]],.],.],.]
=> ([(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]
=> [[[[[.,.],.],.],.],.]
=> ([(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]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(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]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(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]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(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]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(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]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[[[.,[.,.]],.],.]],.]
=> ([(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]
=> [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(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]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(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]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(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]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> ([(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]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of maximal antichains in a poset.
The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001093The detour number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St000259The diameter of a connected graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001120The length of a longest path in a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000050The depth or height of a binary tree. St000863The length of the first row of the shifted shape of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000144The pyramid weight of the Dyck path. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001619The number of non-isomorphic sublattices of a lattice. St001626The number of maximal proper sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001720The minimal length of a chain of small intervals in a lattice. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000503The maximal difference between two elements in a common block. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000553The number of blocks of a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000806The semiperimeter of the associated bargraph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001616The number of neutral elements in a lattice. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001820The size of the image of the pop stack sorting operator. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000245The number of ascents of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001298The number of repeated entries in the Lehmer code of a permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000056The decomposition (or block) number of a permutation. St000240The number of indices that are not small excedances. St000619The number of cyclic descents of a permutation. St000991The number of right-to-left minima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000234The number of global ascents of a permutation. St000691The number of changes of a binary word. St001651The Frankl number of a lattice. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000653The last descent of a permutation. St000141The maximum drop size of a permutation. St000651The maximal size of a rise in a permutation. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001649The length of a longest trail in a graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000652The maximal difference between successive positions of a permutation. St000956The maximal displacement of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001668The number of points of the poset minus the width of the poset. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000765The number of weak records in an integer composition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001645The pebbling number of a connected graph. St000521The number of distinct subtrees of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St001330The hat guessing number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001267The length of the Lyndon factorization of the binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000094The depth of an ordered tree. St000166The depth minus 1 of an ordered tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001391The disjunction number of a graph. St001581The achromatic number of a graph. St001621The number of atoms of a lattice. St001670The connected partition number of a graph. St001927Sparre Andersen's number of positives of a signed permutation. St000075The orbit size of a standard tableau under promotion. St000155The number of exceedances (also excedences) of a permutation. St000273The domination number of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000916The packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001114The number of odd descents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001339The irredundance number of a graph. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001469The holeyness of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001637The number of (upper) dissectors of a poset.
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