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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000143
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> 0
[1,0,1,0]
=> [1]
=> 0
[1,1,0,0]
=> []
=> 0
[1,0,1,0,1,0]
=> [2,1]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> 0
[1,1,0,1,0,0]
=> [1]
=> 0
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St001587
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> [1]
=> 0
[1,1,0,0]
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[1,1,1,0,0,0]
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [4,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [4,1,1,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,1,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,1,1,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,1,1,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [4,1,1,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [4,3]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [4,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1,1,1,1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,1,1,1,1,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,1,1,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [4,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3,1,1,1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,1,1,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [3,1,1]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[]
=> []
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St001685
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 83%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,5,6,4,2,1,7] => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,6,5,4,2,1,7] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,3,5,2,6,1,7] => ? = 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => ? = 0
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => ? = 0
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => ? = 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 0
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,7,6,4,3,2] => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,5,4,3,2] => ? = 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => ? = 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 2
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 2
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => ? = 3
[1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => ? = 1
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => ? = 1
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St001687
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 83%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,6,4] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,4,6,3,2,7] => ? = 4
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 0
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,4,3,2,7] => ? = 4
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 0
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => ? = 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 0
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => ? = 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 0
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => ? = 0
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,6,4,1] => ? = 0
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,3,1] => ? = 0
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => ? = 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => ? = 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St001683
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [.,.]
=> [1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> [2,1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[[.,[[.,[.,.]],.]],.],[.,.]]
=> [3,2,4,1,5,7,6] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[[.,[[[.,.],.],.]],.],[.,.]]
=> [2,3,4,1,5,7,6] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> [2,3,1,4,5,7,6] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [2,6,5,4,3,1,7] => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [3,2,6,5,4,1,7] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> [2,3,6,5,4,1,7] => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [[[.,[[.,.],.]],[.,[.,.]]],.]
=> [2,3,1,6,5,4,7] => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [4,6,5,3,2,1,7] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [2,4,6,5,3,1,7] => ? = 3
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [2,1,4,6,5,3,7] => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [4,3,6,5,2,1,7] => ? = 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [4,3,2,6,5,1,7] => ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> [2,4,3,6,5,1,7] => ? = 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [[[.,[[.,.],[.,.]]],[.,.]],.]
=> [2,4,3,1,6,5,7] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[.,[.,[[[.,.],.],[.,.]]]],.]
=> [3,4,6,5,2,1,7] => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[.,[[.,[[.,.],.]],[.,.]]],.]
=> [3,4,2,6,5,1,7] => ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [[[.,[.,[[.,.],.]]],[.,.]],.]
=> [3,4,2,1,6,5,7] => ? = 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[.,[[[.,[.,.]],.],[.,.]]],.]
=> [3,2,4,6,5,1,7] => ? = 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [[[.,[[.,[.,.]],.]],[.,.]],.]
=> [3,2,4,1,6,5,7] => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[.,[[[[.,.],.],.],[.,.]]],.]
=> [2,3,4,6,5,1,7] => ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [[[.,[[[.,.],.],.]],[.,.]],.]
=> [2,3,4,1,6,5,7] => ? = 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [[[[.,[[.,.],.]],.],[.,.]],.]
=> [2,3,1,4,6,5,7] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => ? = 0
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,5,6,4,3,2,7] => ? = 4
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,[[.,.],[.,[[.,.],.]]]],.]
=> [2,5,6,4,3,1,7] => ? = 3
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [[[.,[.,.]],[.,[[.,.],.]]],.]
=> [2,1,5,6,4,3,7] => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => ? = 2
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> [3,2,5,6,4,1,7] => ? = 2
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [[[.,[.,[.,.]]],[[.,.],.]],.]
=> [3,2,1,5,6,4,7] => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> [2,3,5,6,4,1,7] => ? = 2
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [[[.,[[.,.],.]],[[.,.],.]],.]
=> [2,3,1,5,6,4,7] => ? = 2
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [[[[.,[.,.]],.],[[.,.],.]],.]
=> [2,1,3,5,6,4,7] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => ? = 0
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,5,4,6,3,2,7] => ? = 4
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [[.,[[.,.],[[.,[.,.]],.]]],.]
=> [2,5,4,6,3,1,7] => ? = 3
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000371
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 83%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => [4,2,3,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,4,2,1] => [2,4,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,4,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 0
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,3,1,2] => [3,1,4,2] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [5,2,3,1,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [5,4,3,2,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,5,2,4,1] => [4,5,3,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,5,2,1,4] => [2,5,3,1,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [3,5,4,1,2] => [4,1,3,5,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,5,3,2,1] => [2,3,5,4,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => [5,3,2,4,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [5,3,2,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,3,5,2,1] => [2,5,4,3,1] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [5,4,2,3,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [4,5,2,3,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,3,1,4] => [3,5,2,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,2,4,1,3] => [4,5,2,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,3,4,2,1] => [2,4,5,3,1] => 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,6,4,2] => [4,7,3,6,2,5,1] => [5,6,7,4,3,2,1] => ? = 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [5,7,4,6,3,2,1] => [2,3,6,7,5,4,1] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => [4,7,6,2,5,1,3] => [5,6,2,4,1,7,3] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => [5,7,6,3,4,2,1] => [2,4,6,3,5,7,1] => ? = 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => [5,7,6,3,2,4,1] => [4,3,6,2,5,7,1] => ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => [4,7,6,3,1,2,5] => [3,1,6,4,2,7,5] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => [4,7,6,5,1,2,3] => [5,1,2,4,6,7,3] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => [5,7,6,4,2,3,1] => [3,4,2,6,5,7,1] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [5,7,4,3,2,1,6] => [2,3,4,7,5,1,6] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => [5,7,6,3,2,1,4] => [2,3,6,1,5,7,4] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [5,7,6,4,2,1,3] => [2,4,1,6,5,7,3] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [5,7,6,4,3,1,2] => [3,1,4,6,5,7,2] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [6,5,7,4,3,2,1] => [2,3,4,7,6,5,1] => ? = 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => [7,2,3,4,5,6,1] => [6,7,2,3,4,5,1] => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => [7,2,3,4,5,1,6] => [5,7,2,3,4,1,6] => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => [7,2,3,4,6,1,5] => [6,7,2,3,4,1,5] => ? = 3
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [7,3,4,5,6,2,1] => [2,6,7,3,4,5,1] => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => [7,2,3,4,1,5,6] => [4,7,2,3,1,5,6] => ? = 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => [7,3,4,5,2,6,1] => [6,5,7,3,4,2,1] => ? = 4
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => [7,2,3,5,1,4,6] => [5,7,2,3,1,4,6] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => [7,2,3,6,1,4,5] => [6,7,2,3,1,4,5] => ? = 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,6,4,1] => [7,3,4,6,2,5,1] => [5,6,7,3,4,2,1] => ? = 3
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [7,3,4,5,2,1,6] => [2,5,7,3,4,1,6] => ? = 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,5,7,4,1] => [7,3,4,6,2,1,5] => [2,6,7,3,4,1,5] => ? = 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,5,4,1] => [7,3,4,6,5,1,2] => [5,1,6,3,7,4,2] => ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => [7,4,5,6,3,2,1] => [2,3,6,7,4,5,1] => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => [7,2,3,1,4,5,6] => [3,7,2,1,4,5,6] => ? = 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => [7,3,4,2,5,6,1] => [6,4,7,3,2,5,1] => ? = 4
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [7,3,4,2,5,1,6] => [5,4,7,3,2,1,6] => ? = 3
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,6,7,5,1] => [7,3,4,2,6,1,5] => [6,4,7,3,2,1,5] => ? = 3
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [7,4,5,3,6,2,1] => [2,6,5,7,4,3,1] => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,3,6,7,1] => [7,2,4,1,3,5,6] => [4,7,2,1,3,5,6] => ? = 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,4,5,3,7,6,1] => [7,3,5,2,4,6,1] => [6,5,7,3,2,4,1] => ? = 4
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,3,7,1] => [7,2,5,1,3,4,6] => [5,7,2,1,3,4,6] => ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,3,1] => [7,2,6,1,3,4,5] => [6,7,2,1,3,4,5] => ? = 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,6,3,1] => [7,3,6,2,4,5,1] => [5,6,7,3,2,4,1] => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => [7,3,5,2,4,1,6] => [4,5,7,3,2,1,6] => ? = 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,5,7,3,1] => [7,3,6,2,4,1,5] => [4,6,7,3,2,1,5] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => [7,3,6,2,5,1,4] => [5,6,7,3,2,1,4] => ? = 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,6,5,3,1] => [7,4,6,3,5,2,1] => [2,5,6,7,4,3,1] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => [7,3,4,2,1,5,6] => [2,4,7,3,1,5,6] => ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => [7,4,5,3,2,6,1] => [6,3,5,7,4,2,1] => ? = 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,4,6,3,7,1] => [7,3,5,2,1,4,6] => [2,5,7,3,1,4,6] => ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => [7,3,6,2,1,4,5] => [2,6,7,3,1,4,5] => ? = 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,4,7,6,3,1] => [7,4,6,3,2,5,1] => [5,3,6,7,4,2,1] => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => [7,3,5,4,1,2,6] => [4,1,5,7,3,2,6] => ? = 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => [7,3,6,4,1,2,5] => [4,1,6,7,3,2,5] => ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,4,3,1] => [7,3,6,5,1,2,4] => [5,1,6,3,7,2,4] => ? = 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,6,4,3,1] => [7,4,6,5,2,3,1] => [3,5,2,6,7,4,1] => ? = 2
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation π of {1,…,n}, this is the number of indices j such that there exist indices i,k with i<j<k and π(i)>π(j)>π(k). In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001596
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [.,.]
=> [1,0]
=> [[1],[]]
=> 0
[1,0,1,0]
=> [[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[1,1,0,0]
=> [.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 0
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 0
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 0
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 0
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 0
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5],[]]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[5,5],[1]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [[5,4],[]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3],[]]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[.,.],.],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3],[]]
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1]]
=> ? = 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[[.,[.,.]],.],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [[4,3,3],[1]]
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],[[.,.],.]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> ? = 4
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[.,[[.,.],.]],[[.,.],.]]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[4,3,2],[]]
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2],[]]
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? = 3
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells (i,j) in a skew partition for which the box (i+1,j+1) is also a cell inside the skew partition.
Matching statistic: St000740
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,3,2,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,2,3,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,5,1,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,5,1,4,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5,2,3,4,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5,2,3,4,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5,4,3,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [5,2,3,6,1,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [4,2,6,1,5,3] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [5,2,6,4,1,3] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [4,2,6,1,5,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [3,6,1,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [3,5,1,6,2,4] => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => [4,6,3,1,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [5,6,3,4,1,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => [4,5,6,1,2,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [3,6,1,4,5,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [3,6,1,5,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [4,6,5,1,3,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [3,6,1,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [6,2,3,4,5,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [5,2,3,6,1,4] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [4,2,6,1,5,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [5,2,6,4,1,3] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => [4,2,6,1,5,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => [6,3,2,4,5,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => [5,3,2,6,1,4] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [6,4,3,2,5,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => [6,5,3,4,2,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [5,4,6,2,1,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => [4,6,3,1,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => [5,6,3,4,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => [5,6,4,3,1,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => [4,6,3,1,5,2] => 2 = 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]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 0 + 1
[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]
=> [2,3,4,5,7,1,6] => [6,2,3,4,7,1,5] => ? = 4 + 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]
=> [2,3,4,6,1,7,5] => [5,2,3,7,1,6,4] => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,1,5] => [6,2,3,7,5,1,4] => ? = 3 + 1
[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]
=> [2,3,4,7,1,5,6] => [5,2,3,7,1,6,4] => ? = 3 + 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]
=> [2,3,5,1,6,7,4] => [4,2,7,1,5,6,3] => ? = 2 + 1
[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]
=> [2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,1,7,4] => [5,2,7,4,1,6,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,1,4] => [6,2,7,4,5,1,3] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => [5,2,6,7,1,3,4] => ? = 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]
=> [2,3,6,1,4,7,5] => [4,2,7,1,5,6,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,1,7,4,5] => [4,2,7,1,6,5,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,1,4,5] => [5,2,7,6,1,4,3] => ? = 2 + 1
[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]
=> [2,3,7,1,4,5,6] => [4,2,7,1,5,6,3] => ? = 2 + 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]
=> [2,4,1,5,6,7,3] => [3,7,1,4,5,6,2] => ? = 1 + 1
[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]
=> [2,4,1,5,7,3,6] => [3,6,1,4,7,2,5] => ? = 4 + 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]
=> [2,4,1,6,3,7,5] => [3,5,1,7,2,6,4] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,1,6,7,3,5] => [3,6,1,7,5,2,4] => ? = 3 + 1
[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]
=> [2,4,1,7,3,5,6] => [3,5,1,7,2,6,4] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,1,6,7,3] => [4,7,3,1,5,6,2] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,4,5,1,7,3,6] => [4,6,3,1,7,2,5] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,1,7,3] => [5,7,3,4,1,6,2] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,1,3] => [6,7,3,4,5,1,2] => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [5,6,3,7,1,2,4] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,1,3,7,5] => [4,5,7,1,2,6,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [4,6,7,1,5,2,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [5,6,7,4,1,2,3] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => [4,5,7,1,2,6,3] => ? = 2 + 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]
=> [2,5,1,3,6,7,4] => [3,7,1,4,5,6,2] => ? = 1 + 1
[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]
=> [2,5,1,3,7,4,6] => [3,6,1,4,7,2,5] => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,1,6,3,7,4] => [3,7,1,5,4,6,2] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,1,6,7,3,4] => [3,7,1,6,5,4,2] => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,5,1,7,3,4,6] => [3,6,1,7,5,2,4] => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,1,3,7,4] => [4,7,5,1,3,6,2] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,1,7,3,4] => [4,7,6,1,5,3,2] => ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [5,7,6,4,1,3,2] => ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => [4,6,7,1,5,2,3] => ? = 2 + 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]
=> [2,6,1,3,4,7,5] => [3,7,1,4,5,6,2] => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,6,1,3,7,4,5] => [3,7,1,4,6,5,2] => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,1,7,3,4,5] => [3,7,1,6,5,4,2] => ? = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,1,3,4,5] => [4,7,6,1,5,3,2] => ? = 1 + 1
[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]
=> [2,7,1,3,4,5,6] => [3,7,1,4,5,6,2] => ? = 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]
=> [3,1,4,5,6,7,2] => [7,2,3,4,5,6,1] => ? = 0 + 1
[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]
=> [3,1,4,5,7,2,6] => [6,2,3,4,7,1,5] => ? = 4 + 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]
=> [3,1,4,6,2,7,5] => [5,2,3,7,1,6,4] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,1,4,6,7,2,5] => [6,2,3,7,5,1,4] => ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,4,7,2,5,6] => [5,2,3,7,1,6,4] => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,2,6,7,4] => [4,2,7,1,5,6,3] => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,1,5,6,2,7,4] => [5,2,7,4,1,6,3] => ? = 2 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001816
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,0,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,0,0,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,7,8,12],[2,6,9,10,11,13,14]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,2,3,5,6,10],[4,7,8,9,11,12]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,5,6,7,8,12],[2,4,9,10,11,13,14]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,7,10,12],[2,6,8,9,11,13,14]]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,2,3,5,8,10],[4,6,7,9,11,12]]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,5,6,7,10,12],[2,4,8,9,11,13,14]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,7,8,10,12],[2,4,6,9,11,13,14]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[1,2,3,5,6],[4,7,8,9,10]]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,3,5,6,7,8],[2,4,9,10,11,12]]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,3,5,6,7,10],[2,4,8,9,11,12]]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 0
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,3,4,5,7,8,9,12,16],[2,6,10,11,13,14,15,17,18]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,2,3,5,6,7,10,14],[4,8,9,11,12,13,15,16]]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,3,5,6,7,8,9,12,16],[2,4,10,11,13,14,15,17,18]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,7,10,14],[2,8,9,11,12,13,15,16]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [[1,2,3,4,5,8,12],[6,7,9,10,11,13,14]]
=> ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition λ has dimension equal to the number of standard tableaux of shape λ. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape λ; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St001314
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [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
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[]
=> []
=> [1,0]
=> [1,0]
=> 0
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
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St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn).
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