Identifier
Values
([2],3) => [2] => [1,0,1,0] => [1,1,0,0] => 2
([1,1],3) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([3,1],3) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 4
([2,1,1],3) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 4
([4,2],3) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 6
([3,1,1],3) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 5
([2,2,1,1],3) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 6
([5,3,1],3) => [5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0] => 9
([4,2,1,1],3) => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0] => 8
([3,2,2,1,1],3) => [3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0] => 9
([2],4) => [2] => [1,0,1,0] => [1,1,0,0] => 2
([1,1],4) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([3],4) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
([2,1],4) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([1,1,1],4) => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 3
([4,1],4) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 5
([2,2],4) => [2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 4
([3,1,1],4) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 5
([2,1,1,1],4) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 5
([5,2],4) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 7
([4,1,1],4) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 6
([3,2,1],4) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 6
([3,1,1,1],4) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 6
([2,2,1,1,1],4) => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 7
([6,3],4) => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 9
([5,2,1],4) => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => 8
([4,1,1,1],4) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 7
([4,2,2],4) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 8
([3,3,1,1],4) => [3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 8
([3,2,1,1,1],4) => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => 8
([2,2,2,1,1,1],4) => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0] => 9
([2],5) => [2] => [1,0,1,0] => [1,1,0,0] => 2
([1,1],5) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([3],5) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
([2,1],5) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([1,1,1],5) => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 3
([4],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 4
([3,1],5) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 4
([2,2],5) => [2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 4
([2,1,1],5) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 4
([1,1,1,1],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 4
([5,1],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 6
([3,2],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 5
([4,1,1],5) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 6
([2,2,1],5) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 5
([3,1,1,1],5) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 6
([2,1,1,1,1],5) => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
([6,2],5) => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 8
([5,1,1],5) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 7
([3,3],5) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 6
([4,2,1],5) => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 7
([4,1,1,1],5) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 7
([2,2,2],5) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 6
([3,2,1,1],5) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 7
([3,1,1,1,1],5) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 7
([2,2,1,1,1,1],5) => [2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => 8
([2],6) => [2] => [1,0,1,0] => [1,1,0,0] => 2
([1,1],6) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([3],6) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 3
([2,1],6) => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 3
([1,1,1],6) => [1,1,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 3
([4],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 4
([3,1],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 4
([2,2],6) => [2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 4
([2,1,1],6) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 4
([1,1,1,1],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 4
([5],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 5
([4,1],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 5
([3,2],6) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 5
([3,1,1],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 5
([2,2,1],6) => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 5
([2,1,1,1],6) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 5
([1,1,1,1,1],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5
([6,1],6) => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 7
([4,2],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 6
([5,1,1],6) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 7
([3,3],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 6
([3,2,1],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 6
([4,1,1,1],6) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 7
([2,2,2],6) => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 6
([2,2,1,1],6) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 6
([3,1,1,1,1],6) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 7
([2,1,1,1,1,1],6) => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 7
([4,3],6) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 7
([5,2,1],6) => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => 8
([3,3,1],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 7
([3,2,2],6) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 7
([4,2,1,1],6) => [4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0] => 8
([2,2,2,1],6) => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 7
([3,2,1,1,1],6) => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => 8
([2,2,1,1,1,1,1],6) => [2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0] => 9
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The sum of the heights of the peaks of a Dyck path.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
to partition
Description
Considers a core as a partition.
This embedding is graded and injective but not surjective on $k$-cores for a given parameter $k$, while it is surjective and neither graded nor injective on the collection of all cores.