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Your data matches 376 different statistics following compositions of up to 3 maps.
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Matching statistic: St000150
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000547
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
Description
The number of even non-empty partial sums of an integer partition.
Matching statistic: St000698
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001767
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Matching statistic: St000291
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 10 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 100 => 101 => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 100 => 101 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 100 => 101 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 101 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 100 => 101 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 100 => 101 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 010 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1100 => 1011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 1100 => 1011 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1100 => 1011 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1100 => 1011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1100 => 1011 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1100 => 1011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1000 => 0101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1010 => 1001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 1010 => 1001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 1010 => 1001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1000 => 0101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1100 => 1011 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1100 => 1011 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1100 => 1011 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1100 => 1011 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1100 => 1011 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1000 => 0101 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1001 => 1101 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1000 => 0101 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1010 => 1001 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1010 => 1001 => 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 1000 => 0101 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1100 => 1011 => 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 1100 => 1011 => 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 1100 => 1011 => 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1000 => 0101 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 0101 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 1000 => 0101 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 1000 => 0101 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1000 => 0101 => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0000 => 1010 => 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => 11100 => 10111 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 11100 => 10111 => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => 11100 => 10111 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => 11100 => 10111 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => 11100 => 10111 => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => 11100 => 10111 => 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => 11000 => 01011 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 11010 => 10011 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => 11010 => 10011 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => 11010 => 10011 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => 11000 => 01011 => 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [8,6,2,3,4,5,7,1] => ? => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,3,5,6,1,8] => ? => ? => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,5,7,1,8] => ? => ? => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,3,4,1,6,7] => ? => ? => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [8,6,4,1,2,3,5,7] => ? => ? => ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => 1100000000 => ? => ? = 4
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => 1100000000 => ? => ? = 4
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,2,1,3,4,5,6,7,8,9,10,11] => 11000000000 => ? => ? = 4
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,1,2,4,5,6,7,8,9,10] => 1010000000 => ? => ? = 4
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,8,7,6,5,4,2,1,3,11] => 1101111110 => 1000000011 => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,8,7,6,5,3,2,1,4,11] => 1110111110 => 1000000111 => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [10,9,8,7,5,4,3,2,1,6,11] => 1111101110 => ? => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [10,9,8,6,5,4,3,2,1,7,11] => 1111110110 => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [9,8,10,7,6,5,4,3,2,1,11] => 1111111100 => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [9,8,7,10,6,5,4,3,2,1,11] => 1111111100 => ? => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,8,7,6,5,10,4,3,2,1,11] => 1111111100 => ? => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,8,7,6,5,4,10,3,2,1,11] => 1111111100 => ? => ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,8,7,6,5,4,3,2,10,1,11] => 1111111100 => ? => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,8,7,6,5,4,3,2,1,10,11] => 1111111100 => ? => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [7,6,8,5,9,4,10,3,11,2,12,1] => 11111100000 => ? => ? = 2
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,3,4,5,6,7,8,9,1,10] => 1100000000 => ? => ? = 4
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [10,2,3,4,5,6,7,8,9,11,1] => 1100000000 => ? => ? = 4
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> [11,10,1,2,3,4,5,6,7,8,9] => 1000000001 => 1101010101 => ? = 4
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [10,8,7,6,5,4,3,2,1,9,11] => 1111111100 => ? => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,3,4,5,6,7,8,9,10,1] => 1100000000 => ? => ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => 10101010100 => 10110011001 => ? = 3
[1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,10,11,2,1] => 1100000000 => ? => ? = 4
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,4,5,6,7,8,9,10,1,2] => 1010000000 => ? => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [11,8,9,10,1,2,3,4,5,6,7] => 1000000100 => 1011010101 => ? = 4
Description
The number of descents of a binary word.
Matching statistic: St000390
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [3] => 100 => 101 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1011 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 1001 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 1010 => 1001 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1101 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1001 => 1101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1001 => 1101 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0101 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 10111 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 10011 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 11010 => 10011 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 11011 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 11001 => 11011 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 11001 => 11011 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 01011 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 10001 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 11001 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 10101 => 11001 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01001 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 10110 => 10001 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 10110 => 10001 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 10101 => 11001 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 10101 => 11001 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 10101 => 11001 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 10100 => 01001 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11101 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 10011 => 11101 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 10011 => 11101 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 10011 => 11101 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 10011 => 11101 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 10011 => 11101 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 00101 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 10001 => 00101 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 10001 => 00101 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 10001 => 00101 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 10101 => 3 = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => 111100 => 101111 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => 111010 => 100111 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2] => 111010 => 100111 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => 111001 => 110111 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => 111001 => 110111 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => 111001 => 110111 => 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => 111000 => 010111 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => 110110 => 100011 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => 110101 => 110011 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => 110101 => 110011 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => 110100 => 010011 => 2 = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1101010101 => ? = 4 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => 100000000001 => 110101010101 => ? = 5 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 3 + 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => 1000000101 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => 11000000001 => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,1] => ? => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 3 + 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => 1000001001 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1,1] => 1100000011 => ? => ? = 3 + 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [5,4,1] => 1000010001 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,7,2] => 1100000010 => ? => ? = 3 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 3 + 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [4,5,1] => 1000100001 => ? => ? = 3 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 3 + 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [3,6,1] => 1001000001 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,6,1,1,1] => 1100000111 => ? => ? = 2 + 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,7,1] => 1010000001 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,8,1] => 1100000001 => ? => ? = 3 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,8] => 1010000000 => ? => ? = 4 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? => ? = 3 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => ? => ? = 3 + 1
[1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,7,1] => 1010000001 => ? => ? = 3 + 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,8] => 1110000000 => ? => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ? => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,5] => 1111110000 => ? => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ? => ? => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ? => ? => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> [6,1,1,1,1] => 1000001111 => ? => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,3,1,1,1,1,1,1] => 1100111111 => ? => ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ? => ? => ? = 1 + 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1,1,1,1] => 1100011111 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? => ? => ? => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? => ? => ? => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? => ? => ? => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? => ? => ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? => ? => ? => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? => ? => ? => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => 1111111010 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? => ? => ? => ? = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001584
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 77%●distinct values known / distinct values provided: 60%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 77%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,2,1,3,4,5,6,7,8,9,10,11] => [3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,1,2,4,5,6,7,8,9,10] => [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [10,4,1,2,3,5,6,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,1,5,6,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,1,6,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [4,5,2,3,6,7,8,9,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,10,1] => [4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,5,1,6,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [9,6,1,2,3,4,5,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,6,1,7,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [6,2,3,4,5,7,8,9,10,1] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [5,4,2,3,6,7,8,9,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [4,5,2,3,6,7,8,9,10,1] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [10,6,1,2,3,4,5,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,5,6,1,7,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [9,7,1,2,3,4,5,6,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,6,7,1,8] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [8,2,3,4,5,6,7,9,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,6,7,8,9,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,9,1,2] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,7,8,2,9,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [10,7,1,2,3,4,5,6,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,5,6,7,1,8,9] => [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8,1,2,3,4,5,6,7] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,6,7,8,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
Description
The area statistic between a Dyck path and its bounce path.
The bounce path [[Mp00099]] is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.
Matching statistic: St001115
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 80%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 80%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[[[.,[.,[.,.]]],.],.],.]
=> [3,2,1,4,5,6] => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[[[.,.],[.,[.,.]]],.],.]
=> [4,3,1,2,5,6] => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [[[[.,.],[.,[.,.]]],.],.]
=> [4,3,1,2,5,6] => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [[[[.,.],[.,[.,.]]],.],.]
=> [4,3,1,2,5,6] => 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.]
=> [4,3,2,1,5,6] => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[[[.,[.,.]],.],[.,.]],.]
=> [5,2,1,3,4,6] => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[[.,[.,[.,.]]],[.,.]],.]
=> [5,3,2,1,4,6] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [[[[[.,.],[.,[.,.]]],.],.],.]
=> [4,3,1,2,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1
Description
The number of even descents of a permutation.
Matching statistic: St000264
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> ? = 1 + 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> ? = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5] => ([],5)
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6] => ([],6)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,1,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,2,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001001
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Values
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,5,6,3,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
The following 366 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000454The largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001638The book thickness of a graph. St001654The monophonic hull number of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001116The game chromatic number of a graph. St000260The radius of a connected graph. St000273The domination number of a graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001597The Frobenius rank of a skew partition. St001672The restrained domination number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000258The burning number of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000918The 2-limited packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001306The number of induced paths on four vertices in a graph. St001322The size of a minimal independent dominating set in a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001596The number of two-by-two squares inside a skew partition. St001691The number of kings in a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001093The detour number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000455The second largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000322The skewness of a graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001339The irredundance number of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000068The number of minimal elements in a poset. St000297The number of leading ones in a binary word. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001330The hat guessing number of a graph. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000069The number of maximal elements of a poset. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001827The number of two-component spanning forests of a graph. St001871The number of triconnected components of a graph. St001964The interval resolution global dimension of a poset. St001488The number of corners of a skew partition. St000478Another weight of a partition according to Alladi. St001545The second Elser number of a connected graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000807The sum of the heights of the valleys of the associated bargraph. St001625The Möbius invariant of a lattice. St000451The length of the longest pattern of the form k 1 2. St000741The Colin de Verdière graph invariant. St001130The number of two successive successions in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001765The number of connected components of the friends and strangers graph. St000259The diameter of a connected graph. St001479The number of bridges of a graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001734The lettericity of a graph. St000627The exponent of a binary word. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001344The neighbouring number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000958The number of Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000099The number of valleys of a permutation, including the boundary. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001530The depth of a Dyck path. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001866The nesting alignments of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001928The number of non-overlapping descents in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000889The number of alternating sign matrices with the same antidiagonal sums. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001555The order of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000326The position of the first one in a binary word after appending a 1 at the end. St000805The number of peaks of the associated bargraph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000878The number of ones minus the number of zeros of a binary word. St001568The smallest positive integer that does not appear twice in the partition. St001820The size of the image of the pop stack sorting operator. St000295The length of the border of a binary word. St000124The cardinality of the preimage of the Simion-Schmidt map. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000007The number of saliances of the permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000834The number of right outer peaks of a permutation. St000842The breadth of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000651The maximal size of a rise in a permutation. St000647The number of big descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000862The number of parts of the shifted shape of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001644The dimension of a graph.
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