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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000204
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(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> 0
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 1
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000996
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1
[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] => [3,1,2,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,1,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000155
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [] => ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1
[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] => [3,1,2,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,1,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,4,1,2] => 2
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St001499
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001499: 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
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001499: 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]
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000711
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 83%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [4,2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,4,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,5,1,3,2] => 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [1,5,3,6,4,7,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [1,5,2,6,4,7,3] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [1,5,2,6,3,7,4] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [1,5,3,7,4,6,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [1,5,2,7,4,6,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [1,5,2,7,3,6,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [1,6,3,7,4,5,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [1,6,2,7,4,5,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [1,6,2,7,3,5,4] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [1,6,2,7,3,4,5] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [1,5,3,6,7,4,2] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [1,5,2,6,7,4,3] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [1,5,2,6,7,3,4] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [1,5,3,4,7,6,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [1,5,2,4,7,6,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [1,5,2,3,7,6,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [1,6,3,4,7,5,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [1,6,2,4,7,5,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [1,6,2,3,7,5,4] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [1,6,2,3,7,4,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [1,5,3,4,6,7,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [1,5,2,4,6,7,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [1,5,2,3,6,7,4] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [1,6,3,4,5,7,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [1,6,2,4,5,7,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [1,6,2,3,5,7,4] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [1,6,2,3,4,7,5] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [1,7,2,4,5,6,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [1,7,2,3,5,6,4] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [1,7,2,3,4,6,5] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [1,7,2,3,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [1,5,6,3,4,7,2] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [1,5,6,2,4,7,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [1,5,6,2,3,7,4] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [1,4,7,3,5,6,2] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [1,4,7,2,5,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [1,5,7,3,4,6,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [1,5,7,2,4,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4] => [1,5,7,2,3,6,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => [1,6,7,3,4,5,2] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [1,6,7,2,4,5,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,5,6,1,7,3,4] => [1,6,7,2,3,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [1,6,7,2,3,4,5] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,7,3,4,6] => [1,5,6,3,7,4,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [1,5,6,2,7,4,3] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,5,7,1,3,4,6] => [1,5,6,2,7,3,4] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,6,3,7,4,5] => [1,4,6,7,3,5,2] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,6,1,3,7,4,5] => [1,4,6,7,2,5,3] => ? = 3
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000710
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 83%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [2,4,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,5,3,2,4] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => [1,3,5,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => [1,5,4,2,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,5,2,4,3] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => [2,5,3,1,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => [2,3,5,1,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [2,5,3,4,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => [2,3,5,4,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => [2,5,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => [2,4,5,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => [5,2,1,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => [2,5,1,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => [5,2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,5,1,3,2] => [3,5,4,1,2] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [1,5,3,6,4,7,2] => [1,7,3,5,2,4,6] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [1,4,3,7,5,6,2] => [1,7,3,2,5,6,4] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [1,5,3,7,4,6,2] => [1,7,3,5,2,6,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [1,6,3,7,4,5,2] => [1,7,3,5,6,2,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [1,4,3,6,7,5,2] => [1,7,3,2,6,4,5] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [1,5,3,6,7,4,2] => [1,7,3,6,2,4,5] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [1,5,3,4,7,6,2] => [1,7,3,4,2,6,5] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [1,6,3,4,7,5,2] => [1,7,3,4,6,2,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [1,5,3,4,6,7,2] => [1,7,3,4,2,5,6] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [1,6,3,4,5,7,2] => [1,7,3,4,5,2,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [1,7,3,4,5,6,2] => [1,7,3,4,5,6,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [1,4,6,3,5,7,2] => [1,7,4,2,5,3,6] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [1,4,6,2,5,7,3] => [1,4,7,2,5,3,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [1,5,6,3,4,7,2] => [1,7,4,5,2,3,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [1,5,6,2,4,7,3] => [1,4,7,5,2,3,6] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [1,4,7,3,5,6,2] => [1,7,4,2,5,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [1,4,7,2,5,6,3] => [1,4,7,2,5,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [1,5,7,3,4,6,2] => [1,7,4,5,2,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [1,5,7,2,4,6,3] => [1,4,7,5,2,6,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => [1,6,7,3,4,5,2] => [1,7,4,5,6,2,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [1,6,7,2,4,5,3] => [1,4,7,5,6,2,3] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,7,4,6] => [1,4,6,3,7,5,2] => [1,7,4,2,6,3,5] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6] => [1,4,6,2,7,5,3] => [1,4,7,2,6,3,5] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,7,3,4,6] => [1,5,6,3,7,4,2] => [1,7,4,6,2,3,5] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [1,5,6,2,7,4,3] => [1,4,7,6,2,3,5] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,5,7,1,3,4,6] => [1,5,6,2,7,3,4] => [1,4,6,7,2,3,5] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,7,6] => [1,4,5,3,7,6,2] => [1,7,4,2,3,6,5] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6] => [1,4,5,2,7,6,3] => [1,4,7,2,3,6,5] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [1,3,5,4,7,6,2] => [1,7,2,4,3,6,5] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6] => [1,3,6,4,7,5,2] => [1,7,2,4,6,3,5] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => [1,4,5,3,6,7,2] => [1,7,4,2,3,5,6] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,7] => [1,4,5,2,6,7,3] => [1,4,7,2,3,5,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,3,5,4,6,7] => [1,3,5,4,6,7,2] => [1,7,2,4,3,5,6] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,3,5,6,4,7] => [1,3,6,4,5,7,2] => [1,7,2,4,5,3,6] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,7,4] => [1,3,7,4,5,6,2] => [1,7,2,4,5,6,3] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,6,3,7,4,5] => [1,4,6,7,3,5,2] => [1,7,5,2,6,3,4] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,6,1,3,7,4,5] => [1,4,6,7,2,5,3] => [1,5,7,2,6,3,4] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,6,7,3,4,5] => [1,5,6,7,3,4,2] => [1,7,5,6,2,3,4] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,6,1,7,3,4,5] => [1,5,6,7,2,4,3] => [1,5,7,6,2,3,4] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,6,7,1,3,4,5] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,6,3,4,7,5] => [1,4,5,7,3,6,2] => [1,7,5,2,3,6,4] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,4,5,7,2,6,3] => [1,5,7,2,3,6,4] => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => [1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5] => [1,3,6,7,4,5,2] => [1,7,2,5,6,3,4] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5,7] => [1,4,5,6,3,7,2] => [1,7,5,2,3,4,6] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5,7] => [1,4,5,6,2,7,3] => [1,5,7,2,3,4,6] => ? = 3
Description
The number of big deficiencies of a permutation.
A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$.
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St000619
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 83%
Mp00064: Permutations —reverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0 + 1
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [3,2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [3,2,4,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [2,3,4,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [3,4,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [4,2,3,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,1,4,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [3,4,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [4,3,2,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [3,4,2,5,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [3,5,1,4,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [4,2,3,5,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,4,2] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [4,2,5,1,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [4,3,2,5,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [5,1,4,3,2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,5,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [4,3,5,1,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [5,1,2,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,5,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [3,4,5,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => [4,5,1,2,3] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [3,5,2,4,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [3,4,1,5,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,2,3,4,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [4,1,3,5,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [5,2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,2,4,3,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,3,1,5,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [4,3,5,2,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,5,2] => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,5,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [3,4,5,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,4,5,2,3] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => [4,2,5,3,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [4,5,3,7,1,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [4,6,2,5,3,7,1] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [4,7,1,5,3,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [4,6,2,7,1,5,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [5,3,4,6,2,7,1] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [5,3,4,7,1,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [6,2,4,5,3,7,1] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [7,1,4,5,3,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [6,2,7,1,4,5,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [5,3,6,2,4,7,1] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [5,3,7,1,4,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [7,1,4,6,2,5,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [6,2,5,3,7,1,4] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [5,4,3,6,2,7,1] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [5,4,3,7,1,6,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [6,2,5,4,3,7,1] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [7,1,5,4,3,6,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [6,2,7,1,5,4,3] => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [3,5,4,6,2,7,1] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [3,5,4,7,1,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [5,4,6,2,3,7,1] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [5,4,7,1,3,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [5,4,6,2,7,1,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [6,2,3,5,4,7,1] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [7,1,3,5,4,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [6,2,5,4,7,1,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [7,1,3,6,2,5,4] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [3,4,5,6,2,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,4,5,7,1,6,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [4,5,6,2,3,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [4,5,7,1,3,6,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [4,5,6,2,7,1,3] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [5,6,2,3,4,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [5,7,1,3,4,6,2] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => [5,6,2,4,7,1,3] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => [5,7,1,3,6,2,4] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [6,2,3,4,5,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => [7,1,3,4,5,6,2] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [3,4,7,5,6,2,1] => [6,2,4,5,7,1,3] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [3,4,5,7,6,2,1] => [7,1,3,5,6,2,4] => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [6,2,4,7,1,3,5] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [4,6,3,5,2,7,1] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [4,7,1,6,3,5,2] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [4,5,2,6,3,7,1] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => [4,6,3,7,1,5,2] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => [4,7,1,5,2,6,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [6,3,4,5,2,7,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [7,1,6,3,4,5,2] => ? = 2 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000647
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 83%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,1,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,3,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,2,3,1,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,4,3,1,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,1,3,2,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,1,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,2,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,2,5,1,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,5,1,4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [3,4,2,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,3,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,3,2,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [4,3,6,1,5,2,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => [5,2,4,3,6,1,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [6,1,5,2,4,3,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [5,2,6,1,4,3,7] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => [3,4,5,2,6,1,7] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => [3,4,6,1,5,2,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,1,7] => [3,5,2,4,6,1,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [3,6,1,5,2,4,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => [3,5,2,6,1,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,1,7] => [5,2,3,4,6,1,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [6,1,5,2,3,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => [5,2,3,6,1,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => [6,1,3,5,2,4,7] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,1,7] => [5,3,4,2,6,1,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => [6,1,5,3,4,2,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,1,7] => [4,2,5,3,6,1,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [4,2,6,1,5,3,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,1,7] => [6,1,4,2,5,3,7] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [3,5,4,2,6,1,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [3,6,1,5,4,2,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [3,4,2,5,6,1,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,1,7] => [3,4,2,6,1,5,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => [3,6,1,4,2,5,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,1,7] => [4,2,3,5,6,1,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [4,2,3,6,1,5,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,1,7] => [6,1,4,2,3,5,7] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [4,2,6,1,3,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [6,1,5,4,3,2,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,1,7] => [4,3,2,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [4,3,2,6,1,5,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [6,1,4,3,2,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,1,7] => [3,2,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [3,2,4,6,1,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [3,2,6,1,4,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [6,1,3,2,4,5,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [2,3,4,6,1,5,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [2,3,6,1,4,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => [4,3,6,2,5,1,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,1,2,7] => [4,3,5,1,6,2,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,1,2,7] => [6,2,4,3,5,1,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,1,2,7] => [5,1,6,2,4,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,1,2,7] => [6,2,5,1,4,3,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,1,2,7] => [3,4,6,2,5,1,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,1,2,7] => [3,4,5,1,6,2,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,1,2,7] => [3,6,2,4,5,1,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [3,5,1,6,2,4,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => [3,6,2,5,1,4,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,1,2,7] => [6,2,3,4,5,1,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => [6,2,3,5,1,4,7] => ? = 2
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St000120
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St001960
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,2,1,5,3] => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [4,3,1,2,6,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,6,5,3,4] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [6,5,3,1,2,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,3,4,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [5,3,1,2,4,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,1,2,5,3,6] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [6,3,1,2,4,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [4,1,2,6,3,5] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [6,3,5,1,2,4] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,3,5] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [5,3,1,2,6,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [2,1,5,4,3,6] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,6] => [5,4,3,1,2,6] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [5,4,1,2,3,6] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [2,1,3,6,4,5] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => [6,4,1,2,3,5] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,1,2,3,6,4] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,5,4,3] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,5] => [2,1,3,6,5,4] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [3,1,2,6,5,4] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => [6,5,4,1,2,3] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [3,1,2,4,6,5] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [6,5,1,2,3,4] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [3,1,2,4,5,6] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [4,2,1,3,6,5] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [3,1,6,5,2,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [6,5,2,4,1,3] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => [4,2,1,3,5,6] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
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