Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000329
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2,1] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,1,3,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.