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Your data matches 309 different statistics following compositions of up to 3 maps.
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Matching statistic: St000131
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St000131: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0
[.,[.,.]]
=> 0
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 0
[[.,.],[.,[.,.]]]
=> 0
[[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> 0
[[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> 0
[[.,[[.,.],.]],.]
=> 0
[[[.,.],[.,.]],.]
=> 0
[[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[[[[.,.],.],.],.]]}}} in a binary tree.
[[oeis:A159772]] counts binary trees avoiding this pattern.
Matching statistic: St000126
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
St000126: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00009: Binary trees —left rotate⟶ Binary trees
St000126: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [.,.]
=> 0
[.,[.,.]]
=> [[.,.],.]
=> [.,[.,.]]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> [[.,.],.]
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [.,[[.,.],.]]
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [.,[.,[.,.]]]
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [[[.,.],.],.]
=> 0
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [[.,.],[.,.]]
=> 0
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [.,[[[.,.],.],.]]
=> 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [.,[[.,[.,.]],.]]
=> 0
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [.,[.,[[.,.],.]]]
=> 0
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [.,[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> 0
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> 0
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [[.,[.,[.,.]]],.]
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [[.,.],[[.,.],.]]
=> 0
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [[.,.],[.,[.,.]]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [.,[[[[.,.],.],.],.]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [.,[[[.,.],[.,.]],.]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [.,[[[.,.],.],[.,.]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [.,[.,[[[.,.],.],.]]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [.,[.,[[.,.],[.,.]]]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> 0
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,[.,.]]]]]}}} in a binary tree.
[[oeis:A036766]] counts binary trees avoiding this pattern.
Matching statistic: St000129
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
St000129: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00009: Binary trees —left rotate⟶ Binary trees
St000129: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [.,.]
=> 0
[.,[.,.]]
=> [[.,.],.]
=> [.,[.,.]]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> [[.,.],.]
=> 0
[.,[.,[.,.]]]
=> [[.,.],[.,.]]
=> [[[.,.],.],.]
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [.,[.,[.,.]]]
=> 0
[[.,.],[.,.]]
=> [[[.,.],.],.]
=> [.,[[.,.],.]]
=> 0
[[.,[.,.]],.]
=> [.,[.,[.,.]]]
=> [[.,.],[.,.]]
=> 0
[[[.,.],.],.]
=> [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 0
[.,[.,[.,[.,.]]]]
=> [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 0
[.,[.,[[.,.],.]]]
=> [[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> 0
[.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[.,[[.,[.,.]],.]]
=> [[.,[.,[.,.]]],.]
=> [.,[.,[.,[.,.]]]]
=> 0
[.,[[[.,.],.],.]]
=> [[.,[[.,.],.]],.]
=> [.,[.,[[.,.],.]]]
=> 0
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> 0
[[.,.],[[.,.],.]]
=> [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 0
[[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> [.,[[.,[.,.]],.]]
=> 0
[[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> [.,[[[.,.],.],.]]
=> 0
[[.,[.,[.,.]]],.]
=> [.,[.,[.,[.,.]]]]
=> [[.,.],[.,[.,.]]]
=> 0
[[.,[[.,.],.]],.]
=> [.,[.,[[.,.],.]]]
=> [[.,.],[[.,.],.]]
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],.]
=> [.,[[.,[.,.]],.]]
=> [[.,[.,[.,.]]],.]
=> 0
[[[[.,.],.],.],.]
=> [.,[[[.,.],.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [.,[.,[[.,.],[.,.]]]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [[[.,.],[.,[.,.]]],.]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [[[.,.],[[.,.],.]],.]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[[.,[.,.]],[.,.]],.]
=> [.,[[.,[.,.]],[.,.]]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> 0
[[[.,.],.],[[.,.],.]]
=> [[[[.,.],.],[.,.]],.]
=> [.,[[[.,.],.],[.,.]]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> [.,[[[.,.],[.,.]],.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> [.,[[[[.,.],.],.],.]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[[.,.],.],.]]]}}} in a binary tree.
[[oeis:A159770]] counts binary trees avoiding this pattern.
Matching statistic: St000966
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
Number of peaks minus the global dimension of the corresponding LNakayama algebra.
Matching statistic: St001551
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001551: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001551: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,2] => 0
[[.,.],.]
=> [1,0,1,0]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
Description
The number of restricted non-inversions between exceedances where the rightmost exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001256
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Matching statistic: St000052
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000127
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000127: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000127: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [.,.]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,[[.,.],.]]]]}}} in a binary tree.
[[oeis:A159768]] counts binary trees avoiding this pattern.
Matching statistic: St000132
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [.,.]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A159773]] counts binary trees avoiding this pattern.
Matching statistic: St000232
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [2,1] => {{1,2}}
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => {{1,4},{2,3}}
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => {{1,4,5},{2,3}}
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => {{1,2,5},{3,4}}
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => {{1,6},{2,3,4,5}}
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => {{1,5,6},{2,3,4}}
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => {{1,2,6},{3,4,5}}
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => {{1,6},{2,5},{3,4}}
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => {{1,4,5,6},{2,3}}
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}}
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => {{1,2,5,6},{3,4}}
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}}
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => {{1,2,3,6},{4,5}}
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => {{1,6},{2,3,5},{4}}
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}}
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => {{1,2,6},{3,5},{4}}
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => {{1,2,3,5,6},{4}}
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
=> 0
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
The following 299 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000359The number of occurrences of the pattern 23-1. St000407The number of occurrences of the pattern 2143 in a permutation. St000498The lcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St000666The number of right tethers of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001550The number of inversions between exceedances where the greater exceedance is linked. St001584The area statistic between a Dyck path and its bounce path. St001705The number of occurrences of the pattern 2413 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000253The crossing number of a set partition. St000544The cop number of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000623The number of occurrences of the pattern 52341 in a permutation. St000962The 3-shifted major index of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000264The girth of a graph, which is not a tree. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001545The second Elser number of a connected graph. St000914The sum of the values of the Möbius function of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001271The competition number of a graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001175The size of a partition minus the hook length of the base cell. St001281The normalized isoperimetric number of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001570The minimal number of edges to add to make a graph Hamiltonian. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001549The number of restricted non-inversions between exceedances. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000570The Edelman-Greene number of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001665The number of pure excedances of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001330The hat guessing number of a graph. St000655The length of the minimal rise of a Dyck path. St001720The minimal length of a chain of small intervals in a lattice. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000456The monochromatic index of a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000629The defect of a binary word. St001964The interval resolution global dimension of a poset. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001783The number of odd automorphisms of a graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001895The oddness of a signed permutation. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000287The number of connected components of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001828The Euler characteristic of a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000286The number of connected components of the complement of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001890The maximum magnitude of the Möbius function of a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000068The number of minimal elements in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001592The maximal number of simple paths between any two different vertices of a graph. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000878The number of ones minus the number of zeros of a binary word. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St000297The number of leading ones in a binary word. St000439The position of the first down step of a Dyck path. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000366The number of double descents of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001115The number of even descents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000007The number of saliances of the permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000807The sum of the heights of the valleys of the associated bargraph. St000894The trace of an alternating sign matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000905The number of different multiplicities of parts of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001765The number of connected components of the friends and strangers graph. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000822The Hadwiger number of the graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001651The Frankl number of a lattice. St001927Sparre Andersen's number of positives of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000678The number of up steps after the last double rise of a Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001141The number of occurrences of hills of size 3 in a Dyck path. St000397The Strahler number of a rooted tree. St000675The number of centered multitunnels of a Dyck path. St000268The number of strongly connected orientations of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001277The degeneracy of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001546The number of monomials in the Tutte polynomial of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001139The number of occurrences of hills of size 2 in a Dyck path. St000523The number of 2-protected nodes of a rooted tree. St000700The protection number of an ordered tree. St000981The length of the longest zigzag subpath. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000385The number of vertices with out-degree 1 in a binary tree. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000731The number of double exceedences of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000069The number of maximal elements of a poset. St000237The number of small exceedances.
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