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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St000053
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St001197
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001199
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001499
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001506
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
Description
Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.
Matching statistic: St000010
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [3,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [2,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [3,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [3,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [3,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [2,2]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000011
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000097
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000288
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0]
=> [1,2] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => 111 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => 110 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => 11 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => 1001 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => 1011 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => 1110 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => 110 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => 111 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => 111 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => 1100 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => 111 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => 101 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => 110 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => 11 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => 11 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => 110 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => 11 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => 11 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => 11 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => 11 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => 10001 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => 10011 => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => 1001 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => 1001 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => 10110 => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => 1010 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => 1011 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => 1011 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => 101 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => 11100 => 3 = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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:
St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000806The semiperimeter of the associated bargraph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000306The bounce count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000822The Hadwiger number of the graph. St001202Call 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 special CNakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000767The number of runs in an integer composition. St000260The radius of a connected graph. St000221The number of strong fixed points of a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000455The second largest eigenvalue of a graph if it is integral. St001372The length of a longest cyclic run of ones of a binary word. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St000741The Colin de Verdière graph invariant. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000068The number of minimal elements in a poset.
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