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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1]
=> 1
[1,1] => [1,1,0,0]
=> [2,1] => [2]
=> 2
[1,2] => [1,0,1,0]
=> [1,2] => [1,1]
=> 1
[2,1] => [1,0,1,0]
=> [1,2] => [1,1]
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> [3,2,1] => [2,1]
=> 2
[1,1,2] => [1,1,0,1,0,0]
=> [2,3,1] => [3]
=> 3
[1,2,1] => [1,1,0,1,0,0]
=> [2,3,1] => [3]
=> 3
[2,1,1] => [1,1,0,1,0,0]
=> [2,3,1] => [3]
=> 3
[1,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [2,2]
=> 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> 4
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> 4
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> 4
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> 4
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 3
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000485
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ? = 1
[1,1] => [1,1,0,0]
=> [2,1] => 2
[1,2] => [1,0,1,0]
=> [1,2] => 1
[2,1] => [1,0,1,0]
=> [1,2] => 1
[1,1,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,1,2] => [1,1,0,1,0,0]
=> [2,3,1] => 3
[1,2,1] => [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1,1] => [1,1,0,1,0,0]
=> [2,3,1] => 3
[1,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
Description
The length of the longest cycle of a permutation.
Matching statistic: St001062
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => {{1}}
=> ? = 1
[1,1] => [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2
[1,2] => [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 1
[2,1] => [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 1
[1,1,1] => [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2
[1,1,2] => [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,2,1] => [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[2,1,1] => [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,2,3,4}}
=> 4
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,2,3,4}}
=> 4
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,2,3,4}}
=> 4
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,2,3,4}}
=> 4
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
Description
The maximal size of a block of a set partition.
Matching statistic: St001232
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[2,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,1,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[1,1,2] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,2,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 2
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 3
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[2,4,1,1] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[4,1,1,2] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[4,1,2,1] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[4,2,1,1] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,1,2,3,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,2,4,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,3,2,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,3,4,2] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,4,2,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,4,3,2] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,1,3,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,1,4,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,3,1,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,3,4,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,4,1,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,4,3,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,3,1,2,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,3,1,4,2] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,3,2,1,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,3,2,4,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,3,4,1,2] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,3,4,2,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,4,1,2,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,4,1,3,2] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,4,2,1,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,4,2,3,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,4,3,1,2] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,4,3,2,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,1,1,3,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,1,1,4,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,1,3,1,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,1,3,4,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,1,4,1,3] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,1,4,3,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,3,1,1,4] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,3,1,4,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,3,4,1,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000317
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [(1,2)]
=> [2,1] => 0 = 1 - 1
[1,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0 = 1 - 1
[2,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 1 = 2 - 1
[1,1,2] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 3 - 1
[1,2,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 3 - 1
[2,1,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 2 = 3 - 1
[1,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 1 = 2 - 1
[1,3,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 1 = 2 - 1
[3,1,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 2 - 1
[2,1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 2 - 1
[2,2,1] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[1,3,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[2,1,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[2,3,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[3,1,2] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[3,2,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 1 - 1
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 2 - 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 4 - 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 4 - 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 4 - 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 4 - 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3 - 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3 - 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3 - 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 3 - 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 - 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 - 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 - 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 2 - 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 3 - 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 3 - 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 3 - 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 3 - 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 3 - 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 3 - 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 4 - 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[2,4,1,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[4,1,1,2] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[4,1,2,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[4,2,1,1] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? = 3 - 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 2 - 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 2 - 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 2 - 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 2 - 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 2 - 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 2 - 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 2 - 1
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
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