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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St000645
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ 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
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[[[[.,.],.],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[[[[.,.],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [[[[[.,.],.],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[[[.,.],[.,[.,.]]],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [[[[[.,.],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [[[[[.,.],.],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [[[[.,.],[.,[.,.]]],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [[[[.,.],[.,[.,.]]],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 7
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [[[[[[.,.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 8
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 8
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000874
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000874: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000874: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 6
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,6,5,4,2,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [3,4,6,5,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [3,6,5,2,4,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [4,6,3,2,5,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 6
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 7
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 7
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 7
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 8
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 6
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 6
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 6
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 7
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 8
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 7
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,1,6,4,5] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> 8
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,7,6,3,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [5,7,4,3,6,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,5,7,3,6,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 8
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 8
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [4,3,7,5,6,2,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 8
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [2,5,7,6,4,3,1] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 8
Description
The position of the last double rise in a Dyck path.
If the Dyck path has no double rises, this statistic is $0$.
Matching statistic: St001879
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 8
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 8
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 9
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 8
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000400
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000400: Ordered 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
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[]],[[]],[]]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [[],[],[],[],[],[]]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[],[],[],[[]],[]]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [[],[],[[]],[],[]]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [[],[],[[]],[],[]]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[],[],[[],[]],[]]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [[],[[]],[],[],[]]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [[],[[]],[],[],[]]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [[],[[]],[],[],[]]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[],[[]],[[]],[]]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [[],[[],[]],[],[]]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [[],[[],[]],[],[]]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[],[[],[],[]],[]]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[[]],[],[],[],[]]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[[]],[],[],[],[]]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[[]],[],[],[],[]]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> [[[]],[],[[]],[]]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [[[]],[[]],[],[]]
=> 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [[[]],[[]],[],[]]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [[[]],[[],[]],[]]
=> 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [[[],[]],[],[],[]]
=> 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [[[],[]],[],[],[]]
=> 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [[[],[]],[[]],[]]
=> 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [[[],[],[]],[],[]]
=> 9 = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[],[],[],[],[],[],[]]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[],[],[],[[]],[]]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[],[],[[]],[],[]]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[],[],[[]],[],[]]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[],[],[[],[]],[]]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[],[[]],[],[],[]]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[],[[]],[],[],[]]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[],[[]],[],[],[]]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[],[[]],[[]],[]]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[],[[],[]],[],[]]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[],[[],[]],[],[]]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[],[[],[],[]],[]]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[],[[]],[],[],[],[]]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [[],[[]],[],[[]],[]]
=> 9 = 8 + 1
Description
The path length of an ordered tree.
This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].
Matching statistic: St000438
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000438: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000438: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,6,5,4,2,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [3,4,6,5,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [3,6,5,2,4,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [4,6,3,2,5,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,1,6,4,5] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,7,6,3,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [5,7,4,3,6,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,5,7,3,6,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [4,3,7,5,6,2,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [2,5,7,6,4,3,1] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
Description
The position of the last up step in a Dyck path.
Matching statistic: St000625
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000625: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000625: 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]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 7 = 6 + 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]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [1,3,2,6,5,4] => 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [1,3,2,6,5,4] => 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [1,3,2,6,5,4] => 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0]]
=> [1,3,2,6,5,4] => 9 = 8 + 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]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7] => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7] => 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7] => 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7] => 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,-1,1],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,4,3,7,6,5] => 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,5,6,7] => 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,6,5,7] => 9 = 8 + 1
Description
The sum of the minimal distances to a greater element.
Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is
$$
\sum_{i=1}^n \min_d(\pi_{i-d}>\pi_i\text{ or }\pi_{i+d}>\pi_i)
$$
This statistic appears in [1].
The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently coincides with [2], which satisfies the functional equation
$$
(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x = 0.
$$
Matching statistic: St000979
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000979: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000979: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3,4,5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> {{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> {{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> {{1},{2,4,5},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> {{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> {{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> {{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> {{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> {{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> {{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> {{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5,6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3},{4,6},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4,5,6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> {{1},{2},{3,5},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> {{1},{2},{3,6},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> {{1},{2},{3,5,6},{4},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2},{3,4,5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> {{1},{2},{3,4,6},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2},{3,4,5,6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4},{5,6},{7}}
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 9 = 8 + 1
Description
Half of MacMahon's equal index of a Dyck path.
This is half the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].
Matching statistic: St001034
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St001034: Dyck paths ⟶ 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
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 9 = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 9 = 8 + 1
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St001348
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 9 = 8 + 1
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St001809
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [3,6,5,4,2,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [3,4,6,5,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [3,6,5,2,4,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [4,6,3,2,5,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,1,6,4,5] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> 9 = 8 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [4,7,6,3,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [5,7,4,3,6,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [4,5,7,3,6,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [4,3,7,5,6,2,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 9 = 8 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 10 = 9 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 8 = 7 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [2,5,7,6,4,3,1] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 9 = 8 + 1
Description
The index of the step at the first peak of maximal height in a Dyck path.
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000070The number of antichains in a poset. St000018The number of inversions of a permutation. St000507The number of ascents of a standard tableau. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St000157The number of descents of a standard tableau. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000301The number of facets of the stable set polytope of a graph. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000030The sum of the descent differences of a permutations. St000795The mad of a permutation. St000957The number of Bruhat lower covers of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000984The number of boxes below precisely one peak. St001090The number of pop-stack-sorts needed to sort a permutation. St001176The size of a partition minus its first part. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000147The largest part of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000474Dyson's crank of a partition. St000784The maximum of the length and the largest part of the integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001437The flex of a binary word. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000356The number of occurrences of the pattern 13-2. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000019The cardinality of the support of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St000197The number of entries equal to positive one in the alternating sign matrix. St000145The Dyson rank of a partition. St000327The number of cover relations in a poset. St000288The number of ones in a binary word. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000189The number of elements in the poset. St000463The number of admissible inversions of a permutation. St000422The energy of a graph, if it is integral. St001614The cyclic permutation representation number of a skew partition. St001060The distinguishing index of a graph.
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