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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St000645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: 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
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
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: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 3 = 2 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ?
=> ?
=> ? => ? = 3 + 1
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ?
=> ?
=> ? => ? = 4 + 1
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ?
=> ? => ? = 7 + 1
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> ?
=> ?
=> ? => ? = 2 + 1
[.,[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]]
=> ?
=> ?
=> ? => ? = 3 + 1
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> ?
=> ?
=> ? => ? = 3 + 1
[.,[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]]
=> ?
=> ?
=> ? => ? = 4 + 1
[.,[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]]
=> ?
=> ?
=> ? => ? = 5 + 1
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> ?
=> ?
=> ? => ? = 5 + 1
[.,[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]]
=> ?
=> ?
=> ? => ? = 6 + 1
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> ?
=> ?
=> ? => ? = 6 + 1
[.,[[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]]
=> ?
=> ?
=> ? => ? = 7 + 1
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ?
=> ?
=> ? => ? = 7 + 1
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ?
=> ? => ? = 8 + 1
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ? => ? = 0 + 1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 7 + 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ? => ? = 0 + 1
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ? => ? = 0 + 1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 7 + 1
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 8 + 1
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 9 + 1
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ?
=> ? => ? = 1 + 1
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ?
=> ? => ? = 2 + 1
[[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 5 + 1
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ?
=> ? => ? = 1 + 1
[[[[[[.,.],.],.],.],.],[[.,[.,.]],.]]
=> ?
=> ?
=> ? => ? = 2 + 1
[[[[[.,.],.],.],.],[[[.,[.,.]],.],.]]
=> ?
=> ?
=> ? => ? = 3 + 1
[[[[.,.],.],.],[[[[.,[.,.]],.],.],.]]
=> ?
=> ?
=> ? => ? = 4 + 1
[[[.,.],.],[[[[[.,[.,.]],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 5 + 1
[[[.,.],.],[[[[[[[.,.],.],.],.],.],.],.]]
=> ?
=> ?
=> ? => ? = 6 + 1
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ?
=> ? => ? = 1 + 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11] => ? = 10 + 1
[[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ?
=> ?
=> ? => ? = 8 + 1
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ?
=> ?
=> ? => ? = 2 + 1
[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ?
=> ?
=> ? => ? = 0 + 1
[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ?
=> ?
=> ? => ? = 1 + 1
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ?
=> ?
=> ? => ? = 0 + 1
[.,[[[.,.],[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? => ? = 8 + 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
Description
The first part of an integer composition.
Matching statistic: St000147
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> []
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> []
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 6
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 6
[[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 6
[[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 6
[[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 6
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1]
=> ? = 6
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 6
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1]
=> ? = 6
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 6
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2]
=> ? = 6
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? = 6
[[.,[[[[.,[.,[.,.]]],.],.],.]],.]
=> [4,3,2,5,6,7,1,8] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3]
=> ? = 7
[[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> [3,4,2,5,6,7,1,8] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1]
=> ? = 7
[[.,[[[[[.,.],[.,.]],.],.],.]],.]
=> [4,2,3,5,6,7,1,8] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3]
=> ? = 7
[[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> [3,2,4,5,6,7,1,8] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2]
=> ? = 7
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> ? = 7
[[[.,[.,[[[[.,.],.],.],.]]],.],.]
=> [3,4,5,6,2,1,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> ? = 7
[[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> [3,4,5,2,6,1,7,8] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1]
=> ? = 7
[[[.,[[[.,[.,[.,.]]],.],.]],.],.]
=> [4,3,2,5,6,1,7,8] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3]
=> ? = 7
[[[.,[[[[.,.],[.,.]],.],.]],.],.]
=> [4,2,3,5,6,1,7,8] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3]
=> ? = 7
[[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> [3,2,4,5,6,1,7,8] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2]
=> ? = 7
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [2,3,4,5,6,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> ? = 7
[[[[.,.],[[[[.,.],.],.],.]],.],.]
=> [3,4,5,6,1,2,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> ? = 7
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[.,[.,[[.,.],[.,.]]]],.],.],.]
=> [5,3,4,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[.,[.,[[.,[.,.]],.]]],.],.],.]
=> [4,3,5,2,1,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2]
=> ? = 7
[[[[.,[.,[[[.,.],.],.]]],.],.],.]
=> [3,4,5,2,1,6,7,8] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1]
=> ? = 7
[[[[.,[[.,.],[.,[.,.]]]],.],.],.]
=> [5,4,2,3,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[.,[[.,[.,.]],[.,.]]],.],.],.]
=> [5,3,2,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[.,[[[.,.],.],[.,.]]],.],.],.]
=> [5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[.,[[.,[.,[.,.]]],.]],.],.],.]
=> [4,3,2,5,1,6,7,8] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3]
=> ? = 7
[[[[.,[[[.,.],[.,.]],.]],.],.],.]
=> [4,2,3,5,1,6,7,8] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3]
=> ? = 7
[[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> [3,2,4,5,1,6,7,8] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2]
=> ? = 7
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [2,3,4,5,1,6,7,8] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 7
[[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> [5,4,3,1,2,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> [5,3,4,1,2,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> [4,3,5,1,2,6,7,8] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2]
=> ? = 7
[[[[[.,.],[[[.,.],.],.]],.],.],.]
=> [3,4,5,1,2,6,7,8] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1]
=> ? = 7
[[[[[.,[.,.]],[.,[.,.]]],.],.],.]
=> [5,4,2,1,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [5,4,1,2,3,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [5,3,2,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[.,[[.,.],.]],[.,.]],.],.],.]
=> [5,2,3,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [5,2,1,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 7
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 7
[[[[[.,[.,[[.,.],.]]],.],.],.],.]
=> [3,4,2,1,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1]
=> ? = 7
[[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> [4,2,3,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 7
[[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> [3,2,4,1,5,6,7,8] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2]
=> ? = 7
Description
The largest part of an integer partition.
Matching statistic: St000384
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> []
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> []
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 6
[[.,[[[.,[.,[.,.]]],.],.]],.]
=> [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 6
[[.,[[[.,[[.,.],.]],.],.]],.]
=> [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 6
[[.,[[[[.,.],[.,.]],.],.]],.]
=> [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 6
[[.,[[[[.,[.,.]],.],.],.]],.]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? = 6
[[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 6
[[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 6
[[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 6
[[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 6
[[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 6
[[[.,[[[.,.],[.,.]],.]],.],.]
=> [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 6
[[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 6
[[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 6
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 6
[[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 6
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[.,[.,[[.,.],.]]],.],.],.]
=> [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 6
[[[[.,[[.,.],[.,.]]],.],.],.]
=> [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 6
[[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 6
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 6
[[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 6
[[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 6
[[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 6
[[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 6
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1]
=> ? = 6
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [4,5,6,3,2,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1]
=> ? = 6
[.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [4,5,6,2,3,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1]
=> ? = 6
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [4,5,3,2,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1]
=> ? = 6
[.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> [5,3,4,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1]
=> ? = 6
[.,[[[[[.,.],[.,[.,.]]],.],.],.]]
=> [5,4,2,3,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[[.,.],[[.,.],.]],.],.],.]]
=> [4,5,2,3,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1]
=> ? = 6
[.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [5,3,2,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 6
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1]
=> ? = 6
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 6
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2]
=> ? = 6
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? = 6
[[.,[.,[[[[[.,.],.],.],.],.]]],.]
=> [3,4,5,6,7,2,1,8] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1]
=> ? = 7
[[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [4,5,3,6,2,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1]
=> ? = 7
[[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> [5,4,3,2,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4]
=> ? = 7
[[.,[[[[.,.],[.,[.,.]]],.],.]],.]
=> [5,4,2,3,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4]
=> ? = 7
[[.,[[[[[.,.],.],[.,.]],.],.]],.]
=> [5,2,3,4,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4]
=> ? = 7
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000784
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> []
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> []
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[[[.,[[.,.],.]],.],.],.]
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 5
[[[[[.,[.,.]],.],.],.],.]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 5
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 5
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? = 5
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? = 5
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 5
[[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 6
[[.,[[[.,[.,[.,.]]],.],.]],.]
=> [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 6
[[.,[[[.,[[.,.],.]],.],.]],.]
=> [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 6
[[.,[[[[.,.],[.,.]],.],.]],.]
=> [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 6
[[.,[[[[.,[.,.]],.],.],.]],.]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? = 6
[[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 6
[[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 6
[[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 6
[[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 6
[[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 6
[[[.,[[[.,.],[.,.]],.]],.],.]
=> [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 6
[[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 6
[[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 6
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 6
[[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 6
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[.,[.,[[.,.],.]]],.],.],.]
=> [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 6
[[[[.,[[.,.],[.,.]]],.],.],.]
=> [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 6
[[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 6
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 6
[[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 6
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 6
[[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 6
[[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 6
[[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 6
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2]
=> ? = 5
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1]
=> ? = 5
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1]
=> ? = 6
[.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [4,5,6,3,2,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1]
=> ? = 6
[.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [4,5,6,2,3,7,8,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1]
=> ? = 6
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [4,5,3,2,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1]
=> ? = 6
[.,[[[[.,[[.,.],[.,.]]],.],.],.]]
=> [5,3,4,2,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1]
=> ? = 6
[.,[[[[[.,.],[.,[.,.]]],.],.],.]]
=> [5,4,2,3,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[[.,.],[[.,.],.]],.],.],.]]
=> [4,5,2,3,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1]
=> ? = 6
[.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [5,3,2,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> ? = 6
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3]
=> ? = 6
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1]
=> ? = 6
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St000019
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1] => 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? = 4
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? = 4
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 4
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 4
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? = 5
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 5
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? = 5
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,3,5,7] => ? = 5
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [5,3,4,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? = 5
[.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [6,4,3,1,2,5,7] => ? = 5
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 5
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> [5,4,6,2,3,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? = 5
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 5
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 5
[.,[[[.,[[.,.],[.,.]]],.],.]]
=> [5,3,4,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => ? = 5
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => ? = 5
[.,[[[[.,.],[.,[.,.]]],.],.]]
=> [5,4,2,3,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 5
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 5
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 5
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 5
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 5
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
[[.,.],[.,[[[.,[.,.]],.],.]]]
=> [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [5,6,4,3,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [5,4,6,3,7,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? = 4
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [4,5,6,3,7,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? = 4
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [5,6,3,4,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [4,5,3,6,7,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 4
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [4,3,5,6,7,1,2] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 4
[[.,[.,.]],[[.,[[.,.],.]],.]]
=> [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000141
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1] => 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? = 4
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? = 4
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 4
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 4
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? = 5
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 5
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? = 5
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,3,5,7] => ? = 5
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [5,3,4,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? = 5
[.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [6,4,3,1,2,5,7] => ? = 5
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 5
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> [5,4,6,2,3,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? = 5
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 5
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 5
[.,[[[.,[[.,.],[.,.]]],.],.]]
=> [5,3,4,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => ? = 5
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => ? = 5
[.,[[[[.,.],[.,[.,.]]],.],.]]
=> [5,4,2,3,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 5
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 5
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 5
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 5
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 5
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
[[.,.],[.,[[[.,[.,.]],.],.]]]
=> [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [5,6,4,3,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [5,4,6,3,7,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? = 4
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [4,5,6,3,7,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? = 4
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [5,6,3,4,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4
[[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [4,5,3,6,7,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 4
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [4,3,5,6,7,1,2] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 4
[[.,[.,.]],[[.,[[.,.],.]],.]]
=> [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000054
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4 = 3 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 4 = 3 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? = 4 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? = 4 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 4 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 4 + 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3 + 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3 + 1
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5 + 1
[.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? = 5 + 1
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 5 + 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5 + 1
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? = 5 + 1
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,3,5,7] => ? = 5 + 1
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [5,3,4,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? = 5 + 1
[.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [6,4,3,1,2,5,7] => ? = 5 + 1
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 5 + 1
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5 + 1
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> [5,4,6,2,3,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? = 5 + 1
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? = 5 + 1
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5 + 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5 + 1
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5 + 1
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 5 + 1
[.,[[[.,[[.,.],[.,.]]],.],.]]
=> [5,3,4,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5 + 1
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => ? = 5 + 1
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => ? = 5 + 1
[.,[[[[.,.],[.,[.,.]]],.],.]]
=> [5,4,2,3,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5 + 1
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => ? = 5 + 1
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [5,3,2,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5 + 1
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5 + 1
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 5 + 1
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 5 + 1
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 5 + 1
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 5 + 1
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3 + 1
[[.,.],[.,[[[.,[.,.]],.],.]]]
=> [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => ? = 3 + 1
[[.,.],[[.,[.,[[.,.],.]]],.]]
=> [5,6,4,3,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4 + 1
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [5,4,6,3,7,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ? = 4 + 1
[[.,.],[[.,[[[.,.],.],.]],.]]
=> [4,5,6,3,7,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => ? = 4 + 1
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [5,6,3,4,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 4 + 1
[[.,.],[[[.,[.,[.,.]]],.],.]]
=> [5,4,3,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4 + 1
[[.,.],[[[.,[[.,.],.]],.],.]]
=> [4,5,3,6,7,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => ? = 4 + 1
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4 + 1
[[.,.],[[[[.,[.,.]],.],.],.]]
=> [4,3,5,6,7,1,2] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [5,4,3,1,2,6,7] => ? = 4 + 1
[[.,[.,.]],[[.,[[.,.],.]],.]]
=> [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 3 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000326
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => => ? = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => 1 => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [2,1] => 0 => 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 11 => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 01 => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 10 => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,3,1] => 00 => 3 = 2 + 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 00 => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 111 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,3,4] => 011 => 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 101 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => 001 => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => 001 => 3 = 2 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,4,3] => 110 => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 010 => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,4,2] => 100 => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => 100 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,4,1] => 000 => 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,4,1] => 000 => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,4,3,1] => 000 => 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,4,2,1] => 000 => 4 = 3 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 000 => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,3,4,5] => 0111 => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,2,4,5] => 1011 => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,3,1,4,5] => 0011 => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,1,4,5] => 0011 => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,4,3,5] => 1101 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,1,4,3,5] => 0101 => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,4,2,5] => 1001 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1001 => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => 0001 => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => 0001 => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => 0001 => 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,4,2,1,5] => 0001 => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0001 => 4 = 3 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 1110 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 0110 => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 1010 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 0010 => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,2,1,5,4] => 0010 => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 1100 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [2,1,4,5,3] => 0100 => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1100 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 0100 => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1000 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,3,5,2] => 1000 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1000 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 1000 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0000 => 5 = 4 + 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [8,7,5,4,6,3,2,1] => [1,2,4,5,3,6,7,8] => ? => ? = 0 + 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [8,5,6,4,7,3,2,1] => [1,4,3,5,2,6,7,8] => ? => ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [8,6,4,5,7,3,2,1] => [1,3,5,4,2,6,7,8] => ? => ? = 0 + 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [8,5,4,6,7,3,2,1] => [1,4,5,3,2,6,7,8] => ? => ? = 0 + 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [8,4,5,6,7,3,2,1] => [1,5,4,3,2,6,7,8] => ? => ? = 0 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => [2,3,1,4,6,5,7,8] => ? => ? = 2 + 1
[.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> [8,5,6,7,3,4,2,1] => [1,4,3,2,6,5,7,8] => ? => ? = 0 + 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,3,4,2,1] => [3,4,2,1,6,5,7,8] => ? => ? = 3 + 1
[.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> [7,8,6,4,3,5,2,1] => [2,1,3,5,6,4,7,8] => ? => ? = 1 + 1
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [8,6,7,4,3,5,2,1] => [1,3,2,5,6,4,7,8] => ? => ? = 0 + 1
[.,[.,[[[.,.],.],[.,[[.,.],.]]]]]
=> [7,8,6,3,4,5,2,1] => [2,1,3,6,5,4,7,8] => ? => ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
=> [7,8,5,4,3,6,2,1] => [2,1,4,5,6,3,7,8] => ? => ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [8,7,4,5,3,6,2,1] => [1,2,5,4,6,3,7,8] => ? => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [8,7,5,3,4,6,2,1] => [1,2,4,6,5,3,7,8] => ? => ? = 0 + 1
[.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> [7,8,3,4,5,6,2,1] => [2,1,6,5,4,3,7,8] => ? => ? = 1 + 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => [2,4,3,5,6,1,7,8] => ? => ? = 5 + 1
[.,[.,[[.,[[.,.],[[.,.],.]]],.]]]
=> [6,7,4,5,3,8,2,1] => [3,2,5,4,6,1,7,8] => ? => ? = 5 + 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ? => ? = 5 + 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [7,4,5,6,3,8,2,1] => [2,5,4,3,6,1,7,8] => ? => ? = 5 + 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [7,6,4,3,5,8,2,1] => [2,3,5,6,4,1,7,8] => ? => ? = 5 + 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => [2,3,6,5,4,1,7,8] => ? => ? = 5 + 1
[.,[.,[[[[[.,.],.],.],[.,.]],.]]]
=> [7,3,4,5,6,8,2,1] => [2,6,5,4,3,1,7,8] => ? => ? = 5 + 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [6,5,3,4,7,8,2,1] => [3,4,6,5,2,1,7,8] => ? => ? = 5 + 1
[.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [5,3,4,6,7,8,2,1] => [4,6,5,3,2,1,7,8] => ? => ? = 5 + 1
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,4,2,3,1] => [1,3,4,2,5,7,6,8] => ? => ? = 0 + 1
[.,[[.,.],[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,2,3,1] => [2,3,4,1,5,7,6,8] => ? => ? = 3 + 1
[.,[[.,.],[.,[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,4,2,3,1] => [3,2,4,1,5,7,6,8] => ? => ? = 3 + 1
[.,[[.,.],[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,2,3,1] => [2,4,3,1,5,7,6,8] => ? => ? = 3 + 1
[.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> [8,7,4,5,6,2,3,1] => [1,2,5,4,3,7,6,8] => ? => ? = 0 + 1
[.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> [8,4,5,6,7,2,3,1] => [1,5,4,3,2,7,6,8] => ? => ? = 0 + 1
[.,[[.,[.,.]],[.,[.,[[.,.],.]]]]]
=> [7,8,6,5,3,2,4,1] => [2,1,3,4,6,7,5,8] => ? => ? = 1 + 1
[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
=> [8,6,7,5,3,2,4,1] => [1,3,2,4,6,7,5,8] => ? => ? = 0 + 1
[.,[[.,[.,.]],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,3,2,4,1] => [2,3,1,4,6,7,5,8] => ? => ? = 2 + 1
[.,[[.,[.,.]],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,3,2,4,1] => [3,2,1,4,6,7,5,8] => ? => ? = 2 + 1
[.,[[.,[.,.]],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,3,2,4,1] => [3,2,4,1,6,7,5,8] => ? => ? = 3 + 1
[.,[[[.,.],.],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,2,3,4,1] => [2,3,1,4,7,6,5,8] => ? => ? = 2 + 1
[.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,2,3,4,1] => [1,2,4,3,7,6,5,8] => ? => ? = 0 + 1
[.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,2,3,4,1] => [2,1,4,3,7,6,5,8] => ? => ? = 1 + 1
[.,[[[.,.],.],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,2,3,4,1] => [1,3,4,2,7,6,5,8] => ? => ? = 0 + 1
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,2,3,4,1] => [3,2,4,1,7,6,5,8] => ? => ? = 3 + 1
[.,[[[.,.],.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,2,3,4,1] => [3,4,2,1,7,6,5,8] => ? => ? = 3 + 1
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [7,8,6,4,3,2,5,1] => [2,1,3,5,6,7,4,8] => ? => ? = 1 + 1
[.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
=> [8,6,7,4,3,2,5,1] => [1,3,2,5,6,7,4,8] => ? => ? = 0 + 1
[.,[[.,[.,[.,.]]],[[[.,.],.],.]]]
=> [6,7,8,4,3,2,5,1] => [3,2,1,5,6,7,4,8] => ? => ? = 2 + 1
[.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
=> [7,8,6,3,4,2,5,1] => [2,1,3,6,5,7,4,8] => ? => ? = 1 + 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [7,6,8,4,2,3,5,1] => [2,3,1,5,7,6,4,8] => ? => ? = 2 + 1
[.,[[[.,[.,.]],.],[[.,.],[.,.]]]]
=> [8,6,7,3,2,4,5,1] => [1,3,2,6,7,5,4,8] => ? => ? = 0 + 1
[.,[[[.,[.,.]],.],[[[.,.],.],.]]]
=> [6,7,8,3,2,4,5,1] => [3,2,1,6,7,5,4,8] => ? => ? = 2 + 1
[.,[[[[.,.],.],.],[.,[[.,.],.]]]]
=> [7,8,6,2,3,4,5,1] => [2,1,3,7,6,5,4,8] => ? => ? = 1 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 82%
Values
[.,.]
=> [1,0]
=> [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => [2] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => [1,2] => 2 = 1 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => [3] => 3 = 2 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => [3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,1,2] => 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [1,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,3] => 3 = 2 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [3,1] => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4] => 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4] => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4] => 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4] => 4 = 3 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,1,2] => 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,2,1,1] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,4] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,4] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,4] => 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,4] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,4] => 4 = 3 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,2,1] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [2,3] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [2,3] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1] => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1] => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1] => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1] => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [1,1,1,1,1,1,2] => ? = 1 + 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => [1,1,1,1,1,3] => ? = 2 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [1,1,1,1,1,3] => ? = 2 + 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => [3,1,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [3,1,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,4] => [1,1,1,1,4] => ? = 3 + 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,4] => [1,1,1,1,4] => ? = 3 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,4] => [1,1,1,1,4] => ? = 3 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,4] => [1,1,1,1,4] => ? = 3 + 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [1,1,1,1,4] => ? = 3 + 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,3] => [1,1,1,2,3] => ? = 2 + 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,1,1,2,3] => ? = 2 + 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [4,1,1,1,1] => ? = 0 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [1,1,1,5] => ? = 4 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,3] => [2,1,1,1,3] => ? = 2 + 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,4] => [1,1,2,4] => ? = 3 + 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,2,4] => [1,1,2,4] => ? = 3 + 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,4] => [1,1,2,4] => ? = 3 + 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,2,4] => [1,1,2,4] => ? = 3 + 1
[.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,2,1,1,1] => ? = 0 + 1
[.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,2,1,1,1] => ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
[.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,4,2] => [4,1,1,2] => ? = 1 + 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,6] => [1,1,6] => ? = 5 + 1
[.,[.,[[.,[.,[.,[[.,.],.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,6] => [1,1,6] => ? = 5 + 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,6] => [1,1,6] => ? = 5 + 1
Description
The last part of an integer composition.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000505The biggest entry in the block containing the 1. St000026The position of the first return of a Dyck path. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000653The last descent of a permutation. St000209Maximum difference of elements in cycles. St000501The size of the first part in the decomposition of a permutation. St000795The mad of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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