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Your data matches 143 different statistics following compositions of up to 3 maps.
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Matching statistic: St001067
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 2
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000142
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> 0
[1,1] => [[1,1],[]]
=> [1,1]
=> 0
[2] => [[2],[]]
=> [2]
=> 1
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> 0
[1,2] => [[2,1],[]]
=> [2,1]
=> 1
[2,1] => [[2,2],[1]]
=> [2,2]
=> 2
[3] => [[3],[]]
=> [3]
=> 0
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 1
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 2
[1,3] => [[3,1],[]]
=> [3,1]
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 3
[2,2] => [[3,2],[1]]
=> [3,2]
=> 1
[3,1] => [[3,3],[2]]
=> [3,3]
=> 0
[4] => [[4],[]]
=> [4]
=> 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 0
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> 2
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 3
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> 0
[1,4] => [[4,1],[]]
=> [4,1]
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 4
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 2
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
[2,3] => [[4,2],[1]]
=> [4,2]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 0
[3,2] => [[4,3],[2]]
=> [4,3]
=> 1
[4,1] => [[4,4],[3]]
=> [4,4]
=> 2
[5] => [[5],[]]
=> [5]
=> 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 0
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> 2
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> 3
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> 0
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> 4
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> 2
[1,5] => [[5,1],[]]
=> [5,1]
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> 2
Description
The number of even parts of a partition.
Matching statistic: St000445
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 2
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001189
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001484
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> 0
[1,1] => [1,0,1,0]
=> [1]
=> 1
[2] => [1,1,0,0]
=> []
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
[3] => [1,1,1,0,0,0]
=> []
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 3
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 2
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000148
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> []
=> 0
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 1
[2] => [[2],[]]
=> [2]
=> []
=> 0
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 1
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 0
[3] => [[3],[]]
=> [3]
=> []
=> 0
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 0
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 0
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 1
[4] => [[4],[]]
=> [4]
=> []
=> 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 2
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 1
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 2
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 1
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 0
[5] => [[5],[]]
=> [5]
=> []
=> 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 3
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 3
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 1
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 1
Description
The number of odd parts of a partition.
Matching statistic: St000992
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> []
=> 0
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 1
[2] => [[2],[]]
=> [2]
=> []
=> 0
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 0
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 1
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 2
[3] => [[3],[]]
=> [3]
=> []
=> 0
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 0
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 2
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 3
[4] => [[4],[]]
=> [4]
=> []
=> 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 2
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 1
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 0
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 3
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 4
[5] => [[5],[]]
=> [5]
=> []
=> 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 3
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 3
Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[2] => [1,1,0,0]
=> [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 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: St000776
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[2] => ([],2)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 2 + 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[3] => ([],3)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 4 = 3 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 2 + 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[4] => ([],4)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 5 = 4 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 4 = 3 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 3 = 2 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[5] => ([],5)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> 6 = 5 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 5 = 4 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The maximal multiplicity of an eigenvalue in a graph.
Matching statistic: St000932
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,2] => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,2,1] => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3] => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1] => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,2] => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4] => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1] => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,2] => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,1] => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2] => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,1] => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,3] => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2,2] => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,3,1] => 111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,4] => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,1,2] => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,2,1] => 110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3] => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => 110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2] => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,1] => 110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,5] => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,1,2] => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,2,1] => 101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
The following 133 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001733The number of weak left to right maxima of a Dyck path. St000502The number of successions of a set partitions. St000678The number of up steps after the last double rise of a Dyck path. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St000674The number of hills of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000441The number of successions of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000031The number of cycles in the cycle decomposition of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000153The number of adjacent cycles of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000247The number of singleton blocks of a set partition. St000617The number of global maxima of a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St001249Sum of the odd parts of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001172The number of 1-rises at odd height of a Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001781The interlacing number of a set partition. St001933The largest multiplicity of a part in an integer partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001877Number of indecomposable injective modules with projective dimension 2. St000731The number of double exceedences of a permutation. St001331The size of the minimal feedback vertex set. St000215The number of adjacencies of a permutation, zero appended. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000022The number of fixed points of a 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. St000223The number of nestings in the 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 $3$. St000996The number of exclusive left-to-right maxima of a permutation. St000359The number of occurrences of the pattern 23-1. St000648The number of 2-excedences of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000993The multiplicity of the largest part of an integer partition. St000366The number of double descents of a permutation. St000137The Grundy value of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000117The number of centered tunnels of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000365The number of double ascents of a permutation. St000456The monochromatic index of a connected graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000284The Plancherel distribution on integer partitions. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000732The number of double deficiencies of a permutation. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000864The number of circled entries of the shifted recording tableau of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001461The number of topologically connected components of the chord diagram of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001948The number of augmented double ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001545The second Elser number of a connected graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001615The number of join prime elements of a lattice. St001712The number of natural descents of a standard Young tableau. St001651The Frankl number of a lattice. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001330The hat guessing number of a graph. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001267The length of the Lyndon factorization of the binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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