searching the database
Your data matches 122 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000659
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[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]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 0
[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]
=> 1
[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]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000919
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ 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]
=> [.,[.,.]]
=> 0
[1,1,0,0]
=> [[.,.],.]
=> 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[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]
=> [.,[[.,[.,.]],.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 1
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 2
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 1
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[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]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[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]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 2
[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: St000291
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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%
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] => 11 => 0
[1,1,0,0]
=> [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [2,1] => 101 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[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]
=> [2,1,1,1] => 10111 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 2
[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: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1]
=> 0
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000292
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => 11 => 11 => 0
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 1
[1,1,0,1,0,0]
=> [2,1] => 101 => 101 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 00101 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11001 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => 11 => 00 => 0
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 000 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 001 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 010 => 1
[1,1,0,1,0,0]
=> [2,1] => 101 => 010 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 011 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0001 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 0010 => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1101 => 0010 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1011 => 0100 => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 00001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 00010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 11101 => 00010 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 00101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 11011 => 00100 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 00101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 10111 => 01000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 01001 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 01011 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 01100 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000288
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 10 => 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 010 => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 010 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 110 => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 0110 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 0100 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 0100 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 0100 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 0100 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 1010 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 1010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1100 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 1100 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 1100 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 1100 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 1000 => 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,3,5,8,6] => [1,2,5,3,6,8,4,7] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,3,4,6,7,8] => [1,2,4,5,3,6,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,3,4,6,8,7] => [1,2,4,5,3,6,8,7] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,2,5,3,7,4,6,8] => [1,2,4,6,3,7,5,8] => ? => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,3,4,5,8] => [1,2,5,6,7,3,4,8] => ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ? => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => [1,3,2,4,6,7,5,8] => ? => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => [1,3,2,5,4,6,8,7] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => [1,3,2,7,4,5,8,6] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,2,6,7,8,4,5] => [1,3,2,7,8,4,5,6] => ? => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => [1,3,2,5,6,8,4,7] => ? => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,5,8,6] => [1,4,2,3,6,8,5,7] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,5,2,8,4,6,7] => [1,4,2,6,3,7,8,5] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,2,4,5,7,8] => [1,4,2,5,6,3,7,8] => ? => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,2,4,5,8,7] => [1,4,2,5,6,3,8,7] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => [1,4,2,6,7,3,8,5] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => [1,5,2,6,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,2,3,6,5,7,8] => [1,3,4,2,6,5,7,8] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,4,2,3,6,7,5,8] => [1,3,4,2,7,5,6,8] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,4,2,3,7,5,8,6] => [1,3,4,2,6,8,5,7] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => [1,3,6,2,8,4,5,7] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => [1,4,5,2,3,6,8,7] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,4,5,2,7,3,6,8] => [1,4,6,2,3,7,5,8] => ? => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,4,5,7,2,3,6,8] => [1,5,6,2,3,7,4,8] => ? => ? = 2
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [1,5,6,2,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,5,2,3,7,4,8,6] => [1,3,4,6,2,8,5,7] => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,5,2,3,7,8,4,6] => [1,3,4,7,2,8,5,6] => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,5,2,3,8,4,6,7] => [1,3,4,6,2,7,8,5] => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,2,6,7,3,4,8] => [1,3,6,7,2,4,5,8] => ? => ? = 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => [1,3,6,8,2,4,5,7] => ? => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,2,7,8,3,4,6] => [1,3,6,7,2,8,4,5] => ? => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,5,7,2,3,8,4,6] => [1,4,5,7,2,8,3,6] => ? => ? = 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,4,7,5,8,6] => [2,1,3,4,6,8,5,7] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => [2,1,3,5,4,7,6,8] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,3,5,4,7,8,6] => [2,1,3,5,4,8,6,7] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,6,7,4,8] => [2,1,3,7,4,5,6,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,3,5,6,8,4,7] => [2,1,3,7,4,5,8,6] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => [2,1,3,6,4,7,5,8] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,3,5,7,4,8,6] => [2,1,3,6,4,8,5,7] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,3,5,8,4,6,7] => [2,1,3,6,4,7,8,5] => ? => ? = 3
[1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,4,5,7,8] => [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5,8] => [2,1,3,5,7,4,6,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,3,6,4,8,5,7] => [2,1,3,5,7,4,8,6] => ? => ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,3,6,7,4,5,8] => [2,1,3,6,7,4,5,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,3,6,7,4,8,5] => [2,1,3,6,8,4,5,7] => ? => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,4,5,6,8] => [2,1,3,5,6,7,4,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,4,8,5,6] => [2,1,3,5,7,8,4,6] => ? => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => [2,1,4,3,5,7,6,8] => ? => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,3,6,7,5,8] => [2,1,4,3,7,5,6,8] => ? => ? = 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000389
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 10 => 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 010 => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 101 => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 010 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 001 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 0010 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 1001 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 1001 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 1010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 1001 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 0100 => 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,3,5,8,6] => [1,2,5,3,6,8,4,7] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,3,8,5,6] => [1,2,5,3,7,8,4,6] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,3,4,6,7,8] => [1,2,4,5,3,6,7,8] => ? => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,3,4,6,8,7] => [1,2,4,5,3,6,8,7] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,2,5,3,7,4,6,8] => [1,2,4,6,3,7,5,8] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,8,6] => [1,2,4,6,3,8,5,7] => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,8,4,6,7] => [1,2,4,6,3,7,8,5] => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => [1,2,6,7,3,4,8,5] => ? => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,2,6,7,3,4,5,8] => [1,2,5,6,7,3,4,8] => ? => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => [1,2,5,6,8,3,4,7] => ? => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ? => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,7,5,6,8] => [1,3,2,4,6,7,5,8] => ? => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => [1,3,2,5,4,6,8,7] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ? => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => [1,3,2,7,4,5,8,6] => ? => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,2,6,7,8,4,5] => [1,3,2,7,8,4,5,6] => ? => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => [1,3,2,5,6,8,4,7] => ? => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,4,2,7,5,8,6] => [1,4,2,3,6,8,5,7] => ? => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,5,2,8,4,6,7] => [1,4,2,6,3,7,8,5] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,2,4,5,7,8] => [1,4,2,5,6,3,7,8] => ? => ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,6,2,4,5,8,7] => [1,4,2,5,6,3,8,7] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => [1,4,2,6,7,3,8,5] => ? => ? = 3
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => [1,5,2,6,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,2,3,6,5,7,8] => [1,3,4,2,6,5,7,8] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,4,2,3,6,7,5,8] => [1,3,4,2,7,5,6,8] => ? => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,4,2,3,7,5,8,6] => [1,3,4,2,6,8,5,7] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,4,2,6,3,5,8,7] => [1,3,5,2,6,4,8,7] => ? => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => [1,3,6,2,8,4,5,7] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => [1,4,5,2,3,6,8,7] => ? => ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,4,5,2,7,3,6,8] => [1,4,6,2,3,7,5,8] => ? => ? = 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,4,5,7,2,3,6,8] => [1,5,6,2,3,7,4,8] => ? => ? = 2
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => [1,5,6,2,7,3,8,4] => ? => ? = 3
[1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,5,2,3,6,8,4,7] => [1,3,4,7,2,5,8,6] => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,5,2,3,7,4,8,6] => [1,3,4,6,2,8,5,7] => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,5,2,3,7,8,4,6] => [1,3,4,7,2,8,5,6] => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,5,2,3,8,4,6,7] => [1,3,4,6,2,7,8,5] => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,2,6,7,3,4,8] => [1,3,6,7,2,4,5,8] => ? => ? = 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => [1,3,6,8,2,4,5,7] => ? => ? = 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,5,2,6,8,3,4,7] => [1,3,6,7,2,4,8,5] => ? => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,2,7,8,3,4,6] => [1,3,6,7,2,8,4,5] => ? => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,5,7,2,3,8,4,6] => [1,4,5,7,2,8,3,6] => ? => ? = 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,4,7,5,8,6] => [2,1,3,4,6,8,5,7] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => [2,1,3,5,4,6,7,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => [2,1,3,5,4,7,6,8] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,3,5,4,7,8,6] => [2,1,3,5,4,8,6,7] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,6,7,4,8] => [2,1,3,7,4,5,6,8] => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,3,5,6,8,4,7] => [2,1,3,7,4,5,8,6] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6,8] => [2,1,3,6,4,7,5,8] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,3,5,7,4,8,6] => [2,1,3,6,4,8,5,7] => ? => ? = 3
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,3,5,8,4,6,7] => [2,1,3,6,4,7,8,5] => ? => ? = 3
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000658
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[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
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,3,6,7,4,8] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ?
=> ? = 1
[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
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => ?
=> ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ?
=> ? = 3
[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
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,4,5,6] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ?
=> ? = 3
[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
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,2,8,5,7] => ?
=> ?
=> ? = 3
[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
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ?
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,3,6,7,2,4,5,8] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,4,2,3,6,8,5,7] => ?
=> ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,6,7,3,8,5] => ?
=> ?
=> ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,4,5,2,3,6,8,7] => ?
=> ?
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,4,5,6,2,3,8,7] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,4,5,7,2,8,3,6] => ?
=> ?
=> ? = 2
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,4,6,2,3,7,5,8] => ?
=> ?
=> ? = 2
[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
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,5,6,2,3,7,4,8] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,5,6,2,7,3,4,8] => ?
=> ?
=> ? = 1
[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
[1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,5,7,2,3,4,6,8] => ?
=> ?
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ?
=> ? = 2
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000157
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 1
[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]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ?
=> ? = 1
[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
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,3,7,5,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,2,4,6,3,7,8,5] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,3,8,5,7] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,2,4,6,7,3,5,8] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,3,8,5] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,3,5] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,3,6,7,4,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,3,6,8,4,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,3,7,8,4,6] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,2,5,6,7,3,4,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,2,6,7,3,4,8,5] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ?
=> ? = 2
[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
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,2,5,6,8,4,7] => ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ?
=> ? = 3
[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
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,7,8,5] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,7,4,5,8,6] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,2,7,8,4,5,6] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,5,7,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ? = 3
[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
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,5,7,2,8,6] => ?
=> ? = 2
[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
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,4,6,2,8,5,7] => ?
=> ? = 3
[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
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,2,5,8,6] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,7,2,8,5,6] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4,7,8,6] => ?
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,4,7,8] => ?
=> ? = 2
[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
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,2,6,7,8,4] => ?
=> ? = 2
[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
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,2,4,7,5,8] => ?
=> ? = 2
[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
[1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,3,6,7,2,4,5,8] => ?
=> ? = 2
[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
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,4,2,3,6,8,5,7] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,3,7,8,6] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,2,5,6,3,8,7] => ?
=> ? = 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 $i$.
The following 112 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000507The number of ascents of a standard tableau. St000142The number of even parts of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. 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. St000251The number of nonsingleton blocks of a set partition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001471The magnitude of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000306The bounce count of a Dyck path. 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. St000834The number of right outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000167The number of leaves of an ordered tree. 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. St000386The number of factors DDU in a Dyck path. St000245The number of ascents of a permutation. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St000568The hook number of a binary tree. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000201The number of leaf nodes in a binary tree. St001427The number of descents of a signed permutation. St000647The number of big 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001298The number of repeated entries in the Lehmer code of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000257The number of distinct parts of a partition that occur at least twice. St000779The tier of a permutation. St001469The holeyness of a permutation. St000353The number of inner valleys of a permutation. St000646The number of big ascents of a permutation. St000711The number of big exceedences of 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. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000455The second largest eigenvalue of a graph if it is integral. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001896The number of right descents of a signed permutations. St001597The Frobenius rank of a skew partition. St000660The number of rises of length at least 3 of a Dyck path. St000807The sum of the heights of the valleys of the associated bargraph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001946The number of descents in a parking function. St000805The number of peaks of the associated bargraph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!