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Your data matches 134 different statistics following compositions of up to 3 maps.
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Matching statistic: St000291
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ 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
[1,1] => [2] => 10 => 1
[1,1,1] => [3] => 100 => 1
[1,1,1,1] => [4] => 1000 => 1
[1,1,2] => [2,1] => 101 => 1
[2,1,1] => [1,2] => 110 => 1
[2,2] => [2] => 10 => 1
[1,1,1,1,1] => [5] => 10000 => 1
[1,1,1,2] => [3,1] => 1001 => 1
[1,1,2,1] => [2,1,1] => 1011 => 1
[1,1,3] => [2,1] => 101 => 1
[1,2,1,1] => [1,1,2] => 1110 => 1
[1,2,2] => [1,2] => 110 => 1
[2,1,1,1] => [1,3] => 1100 => 1
[2,2,1] => [2,1] => 101 => 1
[3,1,1] => [1,2] => 110 => 1
[1,1,1,1,1,1] => [6] => 100000 => 1
[1,1,1,1,2] => [4,1] => 10001 => 1
[1,1,1,2,1] => [3,1,1] => 10011 => 1
[1,1,1,3] => [3,1] => 1001 => 1
[1,1,2,1,1] => [2,1,2] => 10110 => 2
[1,1,2,2] => [2,2] => 1010 => 2
[1,1,3,1] => [2,1,1] => 1011 => 1
[1,1,4] => [2,1] => 101 => 1
[1,2,1,1,1] => [1,1,3] => 11100 => 1
[1,2,2,1] => [1,2,1] => 1101 => 1
[1,3,1,1] => [1,1,2] => 1110 => 1
[2,1,1,1,1] => [1,4] => 11000 => 1
[2,1,1,2] => [1,2,1] => 1101 => 1
[2,2,1,1] => [2,2] => 1010 => 2
[2,2,2] => [3] => 100 => 1
[3,1,1,1] => [1,3] => 1100 => 1
[3,3] => [2] => 10 => 1
[4,1,1] => [1,2] => 110 => 1
[1,1,1,1,1,1,1] => [7] => 1000000 => 1
[1,1,1,1,1,2] => [5,1] => 100001 => 1
[1,1,1,1,2,1] => [4,1,1] => 100011 => 1
[1,1,1,1,3] => [4,1] => 10001 => 1
[1,1,1,2,1,1] => [3,1,2] => 100110 => 2
[1,1,1,2,2] => [3,2] => 10010 => 2
[1,1,1,3,1] => [3,1,1] => 10011 => 1
[1,1,1,4] => [3,1] => 1001 => 1
[1,1,2,1,1,1] => [2,1,3] => 101100 => 2
[1,1,2,1,2] => [2,1,1,1] => 10111 => 1
[1,1,2,2,1] => [2,2,1] => 10101 => 2
[1,1,2,3] => [2,1,1] => 1011 => 1
[1,1,3,1,1] => [2,1,2] => 10110 => 2
[1,1,3,2] => [2,1,1] => 1011 => 1
[1,1,4,1] => [2,1,1] => 1011 => 1
[1,1,5] => [2,1] => 101 => 1
[1,2,1,1,1,1] => [1,1,4] => 111000 => 1
Description
The number of descents of a binary word.
Matching statistic: St000659
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => [1,1,0,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,2] => [2] => [1,1,0,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [2] => [1,1,0,0]
=> 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ 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
[1,1] => [2] => [2]
=> 1
[1,1,1] => [3] => [3]
=> 1
[1,1,1,1] => [4] => [4]
=> 1
[1,1,2] => [2,1] => [2,1]
=> 1
[2,1,1] => [1,2] => [2,1]
=> 1
[2,2] => [2] => [2]
=> 1
[1,1,1,1,1] => [5] => [5]
=> 1
[1,1,1,2] => [3,1] => [3,1]
=> 1
[1,1,2,1] => [2,1,1] => [2,1,1]
=> 1
[1,1,3] => [2,1] => [2,1]
=> 1
[1,2,1,1] => [1,1,2] => [2,1,1]
=> 1
[1,2,2] => [1,2] => [2,1]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> 1
[2,2,1] => [2,1] => [2,1]
=> 1
[3,1,1] => [1,2] => [2,1]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> 1
[1,1,1,1,2] => [4,1] => [4,1]
=> 1
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> 1
[1,1,1,3] => [3,1] => [3,1]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> 2
[1,1,2,2] => [2,2] => [2,2]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> 1
[1,1,4] => [2,1] => [2,1]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> 1
[2,1,1,2] => [1,2,1] => [2,1,1]
=> 1
[2,2,1,1] => [2,2] => [2,2]
=> 2
[2,2,2] => [3] => [3]
=> 1
[3,1,1,1] => [1,3] => [3,1]
=> 1
[3,3] => [2] => [2]
=> 1
[4,1,1] => [1,2] => [2,1]
=> 1
[1,1,1,1,1,1,1] => [7] => [7]
=> 1
[1,1,1,1,1,2] => [5,1] => [5,1]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [4,1,1]
=> 1
[1,1,1,1,3] => [4,1] => [4,1]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [3,2,1]
=> 2
[1,1,1,2,2] => [3,2] => [3,2]
=> 2
[1,1,1,3,1] => [3,1,1] => [3,1,1]
=> 1
[1,1,1,4] => [3,1] => [3,1]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [2,2,1]
=> 2
[1,1,2,3] => [2,1,1] => [2,1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [2,2,1]
=> 2
[1,1,3,2] => [2,1,1] => [2,1,1]
=> 1
[1,1,4,1] => [2,1,1] => [2,1,1]
=> 1
[1,1,5] => [2,1] => [2,1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [4,1,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000053
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,3] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000251
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,2] => [2] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[3,3] => [2] => [1,1,0,0]
=> {{1,2}}
=> 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7}}
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ 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
[1,1] => [2] => 10 => 01 => 1
[1,1,1] => [3] => 100 => 001 => 1
[1,1,1,1] => [4] => 1000 => 0001 => 1
[1,1,2] => [2,1] => 101 => 101 => 1
[2,1,1] => [1,2] => 110 => 011 => 1
[2,2] => [2] => 10 => 01 => 1
[1,1,1,1,1] => [5] => 10000 => 00001 => 1
[1,1,1,2] => [3,1] => 1001 => 1001 => 1
[1,1,2,1] => [2,1,1] => 1011 => 1101 => 1
[1,1,3] => [2,1] => 101 => 101 => 1
[1,2,1,1] => [1,1,2] => 1110 => 0111 => 1
[1,2,2] => [1,2] => 110 => 011 => 1
[2,1,1,1] => [1,3] => 1100 => 0011 => 1
[2,2,1] => [2,1] => 101 => 101 => 1
[3,1,1] => [1,2] => 110 => 011 => 1
[1,1,1,1,1,1] => [6] => 100000 => 000001 => 1
[1,1,1,1,2] => [4,1] => 10001 => 10001 => 1
[1,1,1,2,1] => [3,1,1] => 10011 => 11001 => 1
[1,1,1,3] => [3,1] => 1001 => 1001 => 1
[1,1,2,1,1] => [2,1,2] => 10110 => 01101 => 2
[1,1,2,2] => [2,2] => 1010 => 0101 => 2
[1,1,3,1] => [2,1,1] => 1011 => 1101 => 1
[1,1,4] => [2,1] => 101 => 101 => 1
[1,2,1,1,1] => [1,1,3] => 11100 => 00111 => 1
[1,2,2,1] => [1,2,1] => 1101 => 1011 => 1
[1,3,1,1] => [1,1,2] => 1110 => 0111 => 1
[2,1,1,1,1] => [1,4] => 11000 => 00011 => 1
[2,1,1,2] => [1,2,1] => 1101 => 1011 => 1
[2,2,1,1] => [2,2] => 1010 => 0101 => 2
[2,2,2] => [3] => 100 => 001 => 1
[3,1,1,1] => [1,3] => 1100 => 0011 => 1
[3,3] => [2] => 10 => 01 => 1
[4,1,1] => [1,2] => 110 => 011 => 1
[1,1,1,1,1,1,1] => [7] => 1000000 => 0000001 => 1
[1,1,1,1,1,2] => [5,1] => 100001 => 100001 => 1
[1,1,1,1,2,1] => [4,1,1] => 100011 => 110001 => 1
[1,1,1,1,3] => [4,1] => 10001 => 10001 => 1
[1,1,1,2,1,1] => [3,1,2] => 100110 => 011001 => 2
[1,1,1,2,2] => [3,2] => 10010 => 01001 => 2
[1,1,1,3,1] => [3,1,1] => 10011 => 11001 => 1
[1,1,1,4] => [3,1] => 1001 => 1001 => 1
[1,1,2,1,1,1] => [2,1,3] => 101100 => 001101 => 2
[1,1,2,1,2] => [2,1,1,1] => 10111 => 11101 => 1
[1,1,2,2,1] => [2,2,1] => 10101 => 10101 => 2
[1,1,2,3] => [2,1,1] => 1011 => 1101 => 1
[1,1,3,1,1] => [2,1,2] => 10110 => 01101 => 2
[1,1,3,2] => [2,1,1] => 1011 => 1101 => 1
[1,1,4,1] => [2,1,1] => 1011 => 1101 => 1
[1,1,5] => [2,1] => 101 => 101 => 1
[1,2,1,1,1,1] => [1,1,4] => 111000 => 000111 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ 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
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => 10 => 11 => 1
[1,1,1] => [3] => 100 => 011 => 1
[1,1,1,1] => [4] => 1000 => 0011 => 1
[1,1,2] => [2,1] => 101 => 110 => 1
[2,1,1] => [1,2] => 110 => 111 => 1
[2,2] => [2] => 10 => 11 => 1
[1,1,1,1,1] => [5] => 10000 => 00011 => 1
[1,1,1,2] => [3,1] => 1001 => 0110 => 1
[1,1,2,1] => [2,1,1] => 1011 => 1100 => 1
[1,1,3] => [2,1] => 101 => 110 => 1
[1,2,1,1] => [1,1,2] => 1110 => 1111 => 1
[1,2,2] => [1,2] => 110 => 111 => 1
[2,1,1,1] => [1,3] => 1100 => 0111 => 1
[2,2,1] => [2,1] => 101 => 110 => 1
[3,1,1] => [1,2] => 110 => 111 => 1
[1,1,1,1,1,1] => [6] => 100000 => 000011 => 1
[1,1,1,1,2] => [4,1] => 10001 => 00110 => 1
[1,1,1,2,1] => [3,1,1] => 10011 => 01100 => 1
[1,1,1,3] => [3,1] => 1001 => 0110 => 1
[1,1,2,1,1] => [2,1,2] => 10110 => 11011 => 2
[1,1,2,2] => [2,2] => 1010 => 1101 => 2
[1,1,3,1] => [2,1,1] => 1011 => 1100 => 1
[1,1,4] => [2,1] => 101 => 110 => 1
[1,2,1,1,1] => [1,1,3] => 11100 => 01111 => 1
[1,2,2,1] => [1,2,1] => 1101 => 1110 => 1
[1,3,1,1] => [1,1,2] => 1110 => 1111 => 1
[2,1,1,1,1] => [1,4] => 11000 => 00111 => 1
[2,1,1,2] => [1,2,1] => 1101 => 1110 => 1
[2,2,1,1] => [2,2] => 1010 => 1101 => 2
[2,2,2] => [3] => 100 => 011 => 1
[3,1,1,1] => [1,3] => 1100 => 0111 => 1
[3,3] => [2] => 10 => 11 => 1
[4,1,1] => [1,2] => 110 => 111 => 1
[1,1,1,1,1,1,1] => [7] => 1000000 => 0000011 => 1
[1,1,1,1,1,2] => [5,1] => 100001 => 000110 => 1
[1,1,1,1,2,1] => [4,1,1] => 100011 => 001100 => 1
[1,1,1,1,3] => [4,1] => 10001 => 00110 => 1
[1,1,1,2,1,1] => [3,1,2] => 100110 => 011011 => 2
[1,1,1,2,2] => [3,2] => 10010 => 01101 => 2
[1,1,1,3,1] => [3,1,1] => 10011 => 01100 => 1
[1,1,1,4] => [3,1] => 1001 => 0110 => 1
[1,1,2,1,1,1] => [2,1,3] => 101100 => 110011 => 2
[1,1,2,1,2] => [2,1,1,1] => 10111 => 11000 => 1
[1,1,2,2,1] => [2,2,1] => 10101 => 11010 => 2
[1,1,2,3] => [2,1,1] => 1011 => 1100 => 1
[1,1,3,1,1] => [2,1,2] => 10110 => 11011 => 2
[1,1,3,2] => [2,1,1] => 1011 => 1100 => 1
[1,1,4,1] => [2,1,1] => 1011 => 1100 => 1
[1,1,5] => [2,1] => 101 => 110 => 1
[1,2,1,1,1,1] => [1,1,4] => 111000 => 001111 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000919
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
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%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
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%
Values
[1,1] => [2] => [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[2,2] => [2] => [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[3,3] => [2] => [1,1,0,0]
=> [[.,.],.]
=> 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 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: St001011
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,2] => [2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,4] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,2,1] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,2,1,1] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3,1,1,1] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,3] => [2] => [1,1] => [1,0,1,0]
=> 1
[4,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,2,2] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,2,3] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,4,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,5] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001197
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,3] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 124 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001199The dominant 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. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. 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:
St001251The number of parts of a partition that are not congruent 1 modulo 3. 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. St001354The number of series nodes in the modular decomposition of a graph. St000306The bounce count of a Dyck path. 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. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000035The number of left outer peaks of a permutation. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. 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. St000325The width of the tree associated to a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. 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. St000779The tier of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000099The number of valleys of a permutation, including the boundary. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001722The number of minimal chains with small intervals between a binary word and the top element. St001568The smallest positive integer that does not appear twice in the partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000264The girth of a graph, which is not a tree. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000007The number of saliances of the permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000862The number of parts of the shifted shape of a permutation. St000546The number of global descents of a permutation. St000237The number of small exceedances. St000731The number of double exceedences of a permutation. St000842The breadth of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000234The number of global ascents of a permutation. St000534The number of 2-rises of a permutation. St000647The number of big descents of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000451The length of the longest pattern of the form k 1 2. St000787The number of flips required to make a perfect matching noncrossing. St001394The genus of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000441The number of successions of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001381The fertility of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001578The minimal number of edges to add or remove to make a graph a line graph. St001728The number of invisible descents of a permutation. St001964The interval resolution global dimension of a poset.
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