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Your data matches 433 different statistics following compositions of up to 3 maps.
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Matching statistic: St001493
(load all 58 compositions to match this statistic)
(load all 58 compositions to match this statistic)
St001493: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000122
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [5,1,3,2,4] => [[.,[[.,.],[.,.]]],.]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A086581]] counts binary trees avoiding this pattern.
Matching statistic: St000373
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000664
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,2,1] => [2,3,1] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,2,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,3,1] => [2,3,4,1] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => [2,4,1,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,4,2,1] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,4,3,1] => [3,4,1,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => [4,1,3,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => [3,1,4,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,4,3] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,3,5,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => [1,5,3,2,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,4,5,2,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => [1,5,2,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => [1,4,2,5,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,5,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => [2,4,5,1,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [2,5,1,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,2,1] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => [3,5,2,1,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => [4,5,3,2,1] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => [5,4,3,2,1] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => [5,3,2,1,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [4,5,2,1,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => [5,2,1,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => [4,2,1,5,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => [5,2,1,3,4] => 0 = 1 - 1
Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000731
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00277: Permutations —catalanization⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [3,1,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,3,1,2] => [4,2,3,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [1,5,3,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => [2,1,5,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,5,4] => [3,1,2,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [4,1,2,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => [5,3,4,1,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [4,2,3,1,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,3,1,5,2] => [5,2,3,1,4] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,5,4,1,2] => [4,1,3,5,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [5,2,3,1,4] => 0 = 1 - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001549
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [3,1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [3,2,1,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [2,3,1,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,3,2] => [2,4,3,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,1,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => [4,3,1,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [2,3,4,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [4,2,3,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [3,1,4,2,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [3,1,5,4,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [3,2,1,4,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => [3,2,1,5,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [4,3,1,5,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [4,3,2,1,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [5,3,1,4,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [2,3,1,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [2,5,3,1,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,2,4] => [2,4,3,1,5] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => [5,3,2,1,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => [2,5,4,3,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [3,4,1,5,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,4,2,1,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,5,4,2,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [3,2,4,1,5] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [3,2,5,4,1] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,4,3,5,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,4,3,2,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,5,3,4,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [5,1,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [1,2,4,3,5] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [1,5,3,2,4] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [2,5,1,4,3] => 0 = 1 - 1
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001906
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => [3,4,1,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,3,2] => [4,3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,4,1] => [4,3,2,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,4,2,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,4,1] => [4,2,3,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,3,2,1] => [2,3,4,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,2,1] => [2,4,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [3,4,1,2,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [3,5,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,5,4,2] => [3,4,1,5,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,1,3,2,5] => [4,3,1,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,1,3,5,2] => [4,5,1,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,1,5,3,2] => [4,3,5,1,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,1,4,3,2] => [5,3,4,1,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,3,4] => [2,1,5,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2,4,1,5] => [4,3,2,1,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [3,2,4,5,1] => [5,3,2,4,1] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,2,5,4,1] => [4,3,2,5,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,4,2,1,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [5,4,2,3,1] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,5,3,1] => [3,5,4,1,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,2,4,3,1] => [3,4,5,1,2] => 1 = 2 - 1
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Matching statistic: St000836
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000805
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[]]
=> [1] => [1] => 1
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => [1,1] => 1
[1,1,0,0]
=> [[2],[]]
=> [2] => [2] => 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => [1,1,1] => 1
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => [1,2] => 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => [2,1] => 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3] => [3] => 1
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => [2,2] => 1
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => [1,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => [2,1,1] => 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => [1,3] => 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => [2,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => [1,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => [2,2] => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => [3,1] => 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => [4] => 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => [3,2] => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => [2,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => [2,3] => 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => [2,2,2] => 1
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => [3,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => [1,1,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => [2,1,1,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => [1,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1] => [2,1,1,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => [2,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => [3,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => [1,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => [3,1,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => [2,2,1,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => [2,1,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => [2,2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => [3,1,3] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => [1,1,2,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => [1,2,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => [2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => [2,2,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => [1,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => [3,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => [4,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5] => [5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2,4] => [4,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => [2,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [3,3] => [3,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => [2,3,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [3,4] => [4,3] => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> [3,3,3,1] => [3,3,1,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [3,3,4] => [3,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2],[]]
=> [2,2,2,2,2] => [2,2,2,2,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [[3,3,2,2],[]]
=> [3,3,2,2] => [3,2,2,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> [2,3,3,2] => [3,3,2,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2],[]]
=> [4,4,2] => [4,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> [3,3,3,2] => [3,3,2,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [2,2,3,3] => [2,3,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [3,4,3] => [4,3,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> [2,4,4] => [4,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5],[]]
=> [5,5] => [5,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> [3,4,4] => [4,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [1,3,3,3] => [3,3,3,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3],[]]
=> [4,3,3] => [3,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [2,3,3,3] => [3,3,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3],[]]
=> [4,4,3] => [4,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> [3,3,3,3] => [3,3,3,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> [4,4,4] => [4,4,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3}
Description
The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Matching statistic: St000754
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000754: Perfect matchings ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000754: Perfect matchings ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [(1,2)]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3} - 1
Description
The Grundy value for the game of removing nestings in a perfect matching.
A move consists of choosing a nesting, that is two pairs $(a,d)$ and $(b,c)$ with $a < b < c < d$ and replacing them with the two pairs $(a,b)$ and $(c,d)$. The player facing a non-nesting matching looses.
The following 423 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000183The side length of the Durfee square of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000710The number of big deficiencies of a permutation. St000352The Elizalde-Pak rank of a permutation. St000534The number of 2-rises of a permutation. St000662The staircase size of the code of a permutation. St001513The number of nested exceedences of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000640The rank of the largest boolean interval in a poset. St001271The competition number of a graph. St000028The number of stack-sorts needed to sort a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000374The number of exclusive right-to-left minima of a permutation. St000862The number of parts of the shifted shape of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St000451The length of the longest pattern of the form k 1 2. St000648The number of 2-excedences of a permutation. St001394The genus of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001128The exponens consonantiae of a partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000007The number of saliances of the permutation. St000742The number of big ascents of a permutation after prepending zero. St000366The number of double descents of a permutation. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000627The exponent of a binary word. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000864The number of circled entries of the shifted recording tableau of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000542The number of left-to-right-minima of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000764The number of strong records in an integer composition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000781The number of proper colouring schemes of a Ferrers diagram. St001432The order dimension of the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000003The number of standard Young tableaux of the partition. St000053The number of valleys of the Dyck path. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000913The number of ways to refine the partition into singletons. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St000260The radius of a connected graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001933The largest multiplicity of a part in an integer partition. St000872The number of very big descents of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000779The tier of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001964The interval resolution global dimension of a poset. St000516The number of stretching pairs of a permutation. St000883The number of longest increasing subsequences of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001344The neighbouring number of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001162The minimum jump of a permutation. St000058The order of a permutation. St000570The Edelman-Greene number of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000069The number of maximal elements of a poset. St000153The number of adjacent cycles 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 $K[x]/(x^n)$. St001568The smallest positive integer that does not appear twice in the partition. St000842The breadth of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001665The number of pure excedances of a permutation. St000360The number of occurrences of the pattern 32-1. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001728The number of invisible descents of a permutation. St000871The number of very big ascents of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000788The number of nesting-similar perfect matchings of a perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000666The number of right tethers of a permutation. St000768The number of peaks in an integer composition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000031The number of cycles in the cycle decomposition of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000232The number of crossings of a set partition. St000694The number of affine bounded permutations that project to a given permutation. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001590The crossing number of a perfect matching. 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. St000408The number of occurrences of the pattern 4231 in a permutation. St000989The number of final rises of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000068The number of minimal elements in a poset. St000253The crossing number of a set partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000563The number of overlapping pairs of blocks of a set partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000255The number of reduced Kogan faces with the permutation as type. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001621The number of atoms of a lattice. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000456The monochromatic index of a connected graph. St000669The number of permutations obtained by switching ascents or descents of size 2. St000991The number of right-to-left minima of a permutation. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000022The number of fixed points of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. 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. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000223The number of nestings in the permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000665The number of rafts of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001330The hat guessing number of a graph. St000544The cop number of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001569The maximal modular displacement of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. 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 $eA$. St001960The number of descents of a permutation minus one if its first entry is not one. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001399The distinguishing number of a poset. St000632The jump number of the poset. St001875The number of simple modules with projective dimension at most 1. St000629The defect of a binary word. St000256The number of parts from which one can substract 2 and still get an integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001552The number of inversions between excedances and fixed points of a permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000407The number of occurrences of the pattern 2143 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000741The Colin de Verdière graph invariant. St000623The number of occurrences of the pattern 52341 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001130The number of two successive successions in a permutation. St000383The last part of an integer composition. St000668The least common multiple of the parts of the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001867The number of alignments of type EN of a signed permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000900The minimal number of repetitions of a part in an integer composition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001811The Castelnuovo-Mumford regularity of a permutation. St000292The number of ascents of a binary word. St000335The difference of lower and upper interactions. St000486The number of cycles of length at least 3 of a permutation. St000654The first descent of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000902 The minimal number of repetitions of an integer composition. St000983The length of the longest alternating subword. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001889The size of the connectivity set of a signed permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000367The number of simsun double descents of a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000546The number of global descents of a permutation. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000877The depth of the binary word interpreted as a path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000990The first ascent of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001488The number of corners of a skew partition. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000011The number of touch points (or returns) of a Dyck path. St000729The minimal arc length of a set partition. St000889The number of alternating sign matrices with the same antidiagonal sums. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000498The lcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000054The first entry of the permutation. St000635The number of strictly order preserving maps of a poset into itself.
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