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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000651
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 3
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St001933
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 91%
St001933: Integer partitions ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2,1]
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> ? = 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 91%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> []
=> => => ? = 0
[1,0,1,0]
=> [1]
=> 10 => 01 => 1
[1,1,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 011 => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 01011 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 00101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01001 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 011 => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0101011 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 01011001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0101101 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 010111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 01100101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0110011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 01101001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0110101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 011011 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 01110001 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0111001 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 01111 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0010101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 001101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00111 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 010011 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0101001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 01011 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0110001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 000101 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00011 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 00101 => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 01010101011 => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => 01010101101 => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> 1110101010 => 0101010111 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> 101001101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => 01010110101 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => 01011010101 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => 01101010101 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> 10101010100 => 00101010101 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> 10011010100 => 00101011001 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> 10100110100 => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> 10010110100 => 00101101001 => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> 10101001100 => ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> 10100101100 => ? => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> 10010101100 => 00110101001 => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> 10011010010 => ? => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> 10100110010 => ? => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> 10010110010 => ? => ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> 10011001010 => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> 10001101010 => ? => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1]
=> 10100011010 => ? => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1]
=> 1100011010 => 0101100011 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1]
=> 10010011010 => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1]
=> 10001011010 => ? => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1]
=> 10101000110 => ? => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1]
=> 10100100110 => ? => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1]
=> 10010100110 => ? => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,1]
=> 10100010110 => ? => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1]
=> 10010010110 => ? => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1]
=> 10001010110 => ? => ? = 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> 1010100010 => 0100010101 => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> 1010010010 => 0100100101 => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> 1001010010 => 0100101001 => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> 1010001010 => 0101000101 => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> 1001001010 => 0101001001 => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1]
=> 1000101010 => 0101010001 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> 10101010101010 => 01010101010101 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => 0101010101011 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> 111010101010 => 010101010111 => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> 11110101010 => 01010101111 => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ? => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> 1010101101010 => 0101011010101 => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? => ? => ? = 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 91%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> []
=> => => ? = 0
[1,0,1,0]
=> [1]
=> 10 => 01 => 1
[1,1,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 011 => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 01011 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 00101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01001 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 011 => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0101011 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 01011001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0101101 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 010111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 01100101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0110011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 01101001 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0110101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 011011 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 01110001 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0111001 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 01111 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0010101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 001101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00111 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 010011 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0101001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 01011 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0110001 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 000101 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00011 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 00101 => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 01010101011 => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => 01010101101 => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> 1110101010 => 0101010111 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> 101001101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => 01010110101 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => 01011010101 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => 01101010101 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> 10101010100 => 00101010101 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> 10011010100 => 00101011001 => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> 10100110100 => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> 10010110100 => 00101101001 => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> 10101001100 => ? => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> 10100101100 => ? => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> 10010101100 => 00110101001 => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> 10011010010 => ? => ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> 10100110010 => ? => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> 10010110010 => ? => ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> 10011001010 => ? => ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> 10001101010 => ? => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1]
=> 10100011010 => ? => ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1]
=> 1100011010 => 0101100011 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1]
=> 10010011010 => ? => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1]
=> 10001011010 => ? => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1]
=> 10101000110 => ? => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1]
=> 10100100110 => ? => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1]
=> 10010100110 => ? => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,1]
=> 10100010110 => ? => ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1]
=> 10010010110 => ? => ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1]
=> 10001010110 => ? => ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> 1010101000 => 0001010101 => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> 1010100100 => 0010010101 => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> 1010010100 => 0010100101 => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> 1001010100 => 0010101001 => ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> 1010100010 => 0100010101 => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> 1010010010 => 0100100101 => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> 1001010010 => 0100101001 => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> 1010001010 => 0101000101 => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> 1001001010 => 0101001001 => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1]
=> 1000101010 => 0101010001 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> 10101010101010 => 01010101010101 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => 0101010101011 => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> 111010101010 => 010101010111 => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> 11110101010 => 01010101111 => ? = 4
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001498
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 55%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 55%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> ? = 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,1,1,2,1] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,1,1,2,1] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,2,1,1] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,1,2,1,1] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,1,2,1,1] => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,4,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,4,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,5,1] => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000845
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 64%
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => ([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,6,5] => ([(0,6),(1,2),(2,6),(6,3),(6,4),(6,5)],7)
=> ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6] => ([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => ([(0,5),(0,6),(1,3),(3,5),(3,6),(4,2),(6,4)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => ([(0,5),(0,6),(1,4),(4,5),(4,6),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => ([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => ([(0,5),(0,6),(1,4),(3,5),(3,6),(4,3),(6,2)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => ([(0,2),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,4,1,6,7] => ([(0,6),(1,5),(2,6),(3,6),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,4,6,1,7] => ([(0,6),(1,4),(2,5),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => ([(1,5),(2,6),(3,6),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,6,4,5,1,7] => ([(0,6),(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,6,4,5,7,1] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => ([(0,6),(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => ([(0,6),(1,3),(1,4),(3,5),(4,5),(5,6),(6,2)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ([(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,3,4,1,6,7] => ([(0,6),(1,3),(1,5),(2,6),(3,6),(5,2),(6,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,3,4,6,1,7] => ([(0,6),(1,3),(1,4),(2,5),(3,5),(4,2),(5,6)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,6,3,4,5,1,7] => ([(0,6),(1,3),(1,5),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,6,3,4,5,7,1] => ([(1,3),(1,5),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,4,2,5,6,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [7,3,4,5,2,6,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [7,5,3,4,6,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 2
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 64%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [2,1] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,1,2] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,1,2] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,1,2] => [1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,1,2] => [1,3,4,5,2,7,6] => ([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,1,2] => [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,1,3] => [1,3,2,4,6,7,5] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [2,3,1,4,6,7,5] => ([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,1,3] => [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,1,3] => [1,3,4,2,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [2,3,4,1,6,7,5] => ([(0,5),(0,6),(1,4),(3,5),(3,6),(4,3),(6,2)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,1,4] => [1,3,2,5,6,7,4] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,1,4] => [2,3,1,5,6,7,4] => ([(0,5),(0,6),(1,3),(3,5),(3,6),(4,2),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => [1,4,3,5,6,7,2] => ([(0,2),(0,3),(0,4),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => [2,4,3,5,6,7,1] => ([(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => [1,2,5,4,6,7,3] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => [2,1,5,4,6,7,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(3,2),(5,3),(6,3)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,1,6] => [1,3,5,4,6,7,2] => ([(0,4),(0,5),(1,6),(2,6),(5,1),(5,2),(6,3)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => [2,3,5,4,6,7,1] => ([(1,5),(2,6),(3,6),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,1,5] => [1,2,4,6,5,7,3] => ([(0,5),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,1,6] => [1,3,4,6,5,7,2] => ([(0,3),(0,4),(1,6),(2,6),(4,5),(5,1),(5,2)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,1,5] => [1,2,5,6,4,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,4,1,6] => [1,3,5,6,4,7,2] => ([(0,3),(0,5),(1,6),(2,6),(4,2),(5,1),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,1,7] => [2,3,5,6,4,7,1] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,1,6] => [1,4,5,6,3,7,2] => ([(0,2),(0,3),(0,5),(1,6),(3,6),(4,1),(5,4)],7)
=> ? = 2
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [2,4,5,6,3,7,1] => ([(1,3),(1,5),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => [2,3,4,6,7,5,1] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => [2,3,5,6,7,4,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => [2,4,5,6,7,3,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => [2,3,5,7,6,4,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [6,4,3,1,2,5,7] => [2,4,5,7,6,3,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => [2,3,6,7,5,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,4,2,1,3,6,7] => [3,4,6,7,5,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000757
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 82%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [4,2,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [4,3,1,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [4,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [5,3,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [4,3,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [3,3,3,2,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [5,4,2,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [4,4,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [5,3,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [4,3,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [5,2,2,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [5,4,3,1,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [4,4,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [5,3,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [4,3,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [5,4,2,1,1] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [4,4,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [5,3,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [5,4,1,1,1] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [5,2,1,1,1] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [5,4,3,2] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [4,4,3,2] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [5,3,3,2] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [4,3,3,2] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [5,4,2,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [4,4,2,2] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [5,3,2,2] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [5,4,3,1] => ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [4,4,3,1] => ? = 2
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
Matching statistic: St000899
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 82%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [4,2,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [4,3,1,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [4,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [5,3,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [4,3,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [3,3,3,2,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [5,4,2,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [4,4,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [5,3,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [4,3,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [5,2,2,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [5,4,3,1,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [4,4,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [5,3,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [4,3,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [5,4,2,1,1] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [4,4,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [5,3,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [5,4,1,1,1] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [5,2,1,1,1] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [5,4,3,2] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [4,4,3,2] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [5,3,3,2] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [4,3,3,2] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [5,4,2,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [4,4,2,2] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [5,3,2,2] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [5,4,3,1] => ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [4,4,3,1] => ? = 2
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000904
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000904: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 82%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000904: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 82%
Values
[1,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [4,2,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [4,3,1,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [0] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [4,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [5,3,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [4,3,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [3,3,3,2,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [5,4,2,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [4,4,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [5,3,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [4,3,2,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [5,2,2,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [5,4,3,1,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [4,4,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [5,3,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [4,3,3,1,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [5,4,2,1,1] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [4,4,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [5,3,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [5,4,1,1,1] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [5,2,1,1,1] => ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [5,4,3,2] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [4,4,3,2] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [5,3,3,2] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [4,3,3,2] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [5,4,2,2] => ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [4,4,2,2] => ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [5,3,2,2] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
=> [5,4,3,1] => ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [4,4,3,1] => ? = 2
Description
The maximal number of repetitions of an integer composition.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001235The global dimension of the corresponding Comp-Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
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