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Your data matches 85 different statistics following compositions of up to 3 maps.
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Matching statistic: St000919
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000919: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[.,.],.]
=> 1
[1,1,0,0]
=> [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 1
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 2
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000659
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000291
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 101 => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 10001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 10001 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 10010 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 10001 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 10010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
Description
The number of descents of a binary word.
Matching statistic: St000288
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 10 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 100 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 110 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 1000 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 1100 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1000 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 1100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1010 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 1100 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 1100 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 1010 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0100 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0110 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0100 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 1000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> [4,6,1,2,3,7,8,5] => ? => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,1,6,7,2,4,8,5] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [4,1,5,7,2,3,8,6] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,8,6] => ? => ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,6,1,2,7,3,8,5] => ? => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,8,5] => ? => ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ? => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,1,2,6,7,4,5,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8] => ? => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ? => ? = 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,4,6,5,7,8] => ? => ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [3,6,1,7,2,4,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5,7,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,1,6,7,3,4,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,4,7,1,2,5,6,8] => ? => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,4,6,1,2,5,7,8] => ? => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,4,5,1,2,7,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [3,6,1,2,7,4,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,6,3,4,7,5,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,4,6,1,7,2,5,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => ? => ? = 3
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,6,7,4,8] => ? => ? = 2
[1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,5,1,3,4,7,6,8] => ? => ? = 2
[1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [2,5,7,1,3,4,6,8] => ? => ? = 1
[1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [2,5,6,1,3,4,7,8] => ? => ? = 1
[1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,3,7,5,6,8] => ? => ? = 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [2,5,6,7,1,3,4,8] => ? => ? = 1
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [2,4,1,3,6,7,5,8] => ? => ? = 2
[1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,4,1,3,5,7,6,8] => ? => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000157
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,7,1,3,4,8,5,6] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [2,5,7,1,3,8,4,6] => ?
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [2,5,6,1,8,3,4,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,8,4,6] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [2,4,7,1,3,8,5,6] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [2,4,6,7,1,8,3,5] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,4,5,8,1,3,6,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [2,4,5,7,1,8,3,6] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,8,4] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [2,4,7,1,3,5,8,6] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [2,4,5,7,1,3,8,6] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,8,4,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,7,1,8,4,5,6] => ?
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,1,4,5,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,1,4,8,5,6] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,3,5,7,1,8,4,6] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,6,1,3,7,8,5] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,6,7,1,4,8,5] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,8,5] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,5,7,1,4,8,6] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,4,1,5,7,8,3,6] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,8,4,6] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,3,6,7,1,4,5,8] => ?
=> ? = 2
[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,3,6,1,7,4,5,8] => ?
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,6,1,4,7,5,8] => ?
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4,8] => ?
=> ? = 2
[1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,3,5,1,7,4,6,8] => ?
=> ? = 3
[1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,3,5,1,6,7,4,8] => ?
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ?
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,5,6,2,8,3,7] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ?
=> ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ? = 2
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 2
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,6,7,8,2,5] => ?
=> ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,7,8,5] => ?
=> ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,2,4,7,8,5] => ?
=> ? = 2
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,5,6,2,7,8,4] => ?
=> ? = 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000389
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 0100 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 0100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 1010 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 1010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1001 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 1001 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> [4,6,1,2,3,7,8,5] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ? => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,1,6,7,2,4,8,5] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [4,6,1,7,2,3,8,5] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [4,1,5,7,2,3,8,6] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,8,6] => ? => ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,6,1,2,7,3,8,5] => ? => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,8,5] => ? => ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 2
[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ? => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,1,2,6,7,4,5,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8] => ? => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ? => ? = 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,4,6,5,7,8] => ? => ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [3,6,1,7,2,4,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5,7,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,1,6,7,3,4,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,4,7,1,2,5,6,8] => ? => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,4,6,1,2,5,7,8] => ? => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,4,5,1,2,7,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [3,6,1,2,7,4,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,6,3,4,7,5,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,4,6,1,7,2,5,8] => ? => ? = 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 0100 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 0100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 1010 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 1010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 1001 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 1001 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 0101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> [4,6,1,2,3,7,8,5] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [4,5,1,2,7,3,8,6] => ? => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [3,1,6,7,2,4,8,5] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [4,6,1,7,2,3,8,5] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [4,1,5,7,2,3,8,6] => ? => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,8,6] => ? => ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,6,1,2,7,3,8,5] => ? => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,8,5] => ? => ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 2
[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,2,4,8,5] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ? => ? = 2
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,4,6,2,7,8,5] => ? => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,1,2,6,7,4,5,8] => ? => ? = 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8] => ? => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ? => ? = 1
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,4,6,5,7,8] => ? => ? = 2
[1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [3,6,1,7,2,4,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5,7,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,1,6,7,3,4,5,8] => ? => ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => ? => ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,4,7,1,2,5,6,8] => ? => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,4,6,1,2,5,7,8] => ? => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,4,5,1,2,7,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,4,7,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [3,6,1,2,7,4,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,6,3,4,7,5,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5,8] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? = 3
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,4,6,1,7,2,5,8] => ? => ? = 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000142
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,4,6,1,5,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2,3,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,5,1,3,4,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,5,1,4,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [2,4,5,1,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> [4,1,5,2,3,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,4,1,5,2,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,1,2,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [4,1,2,5,3,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,4,2,5,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,7,4,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [2,4,7,1,3,5,6,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,4,7,1,2,5,6,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,4,7,2,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,4,1,7,2,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,3,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,7,3,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [2,5,7,1,3,4,6,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,5,1,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,4,5,7,1,3,6,8] => ?
=> ? = 1
Description
The number of even parts of a partition.
Matching statistic: St000566
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,4,6,1,5,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2,3,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,5,1,3,4,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,5,1,4,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [2,4,5,1,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> [4,1,5,2,3,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,4,1,5,2,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,1,2,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [4,1,2,5,3,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,4,2,5,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,7,4,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [2,4,7,1,3,5,6,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,4,7,1,2,5,6,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,4,7,2,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,4,1,7,2,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,3,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,7,3,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [2,5,7,1,3,4,6,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,5,1,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,4,5,7,1,3,6,8] => ?
=> ? = 1
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St001251
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,4,5,8,7] => ?
=> ? = 3
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,6,4,5,8,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [2,3,4,6,1,5,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2,6,3,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2,3,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,5,1,3,4,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,5,1,4,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [2,4,5,1,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> [4,1,5,2,3,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,4,1,5,2,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,1,2,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [4,1,2,5,3,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,2,5,6,3,4,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,6,4,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,4,2,5,6,8,7] => ?
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6,8,7] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,7,4,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [2,4,7,1,3,5,6,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,4,7,1,2,5,6,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,4,7,2,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,4,1,7,2,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,3,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ? = 3
[1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,7,3,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,7,5,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [2,5,7,1,3,4,6,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [5,1,2,7,3,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,5,1,3,4,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,1,3,7,4,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,7,4,6,8] => ?
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 3
[1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [2,4,5,7,1,3,6,8] => ?
=> ? = 1
Description
The number of parts of a partition that are not congruent 1 modulo 3.
The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000386The number of factors DDU in a Dyck path. St001840The number of descents of a set partition. St000779The tier of a permutation. St000251The number of nonsingleton blocks of a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000646The number of big ascents of a permutation. St000834The number of right outer peaks of a permutation. St000703The number of deficiencies of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000035The number of left outer peaks of a permutation. St000662The staircase size of the code of a permutation. St000201The number of leaf nodes in a binary tree. St000245The number of ascents of a permutation. St001427The number of descents of a signed permutation. St001083The number of boxed occurrences of 132 in a permutation. St000021The number of descents of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St000257The number of distinct parts of a partition that occur at least twice. St000647The number of big descents of a permutation. St000353The number of inner valleys of a permutation. St000619The number of cyclic descents of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001469The holeyness of a permutation. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001689The number of celebrities in a graph. St001728The number of invisible descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000256The number of parts from which one can substract 2 and still get an integer partition. St001896The number of right descents of a signed permutations. St001330The hat guessing number of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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