Identifier
Values
([],1) => [2] => [1,0,1,0] => [1,1,0,0] => 2
([],2) => [2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 3
([(0,1)],2) => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
([],3) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 3
([(1,2)],3) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 5
([(0,1),(0,2)],3) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 4
([(0,2),(2,1)],3) => [4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 3
([(0,2),(1,2)],3) => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 4
([(0,1),(0,2),(0,3)],4) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 4
([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 4
([(1,2),(2,3)],4) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 5
([(0,3),(3,1),(3,2)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 4
([(0,3),(1,3),(3,2)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 4
([(0,3),(1,3),(2,3)],4) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 4
([(0,3),(1,2)],4) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,3),(1,2),(1,3)],4) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 4
([(0,2),(0,3),(1,2),(1,3)],4) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 5
([(0,3),(2,1),(3,2)],4) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 5
([(0,3),(0,4),(3,2),(4,1)],5) => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 5
([(0,4),(1,4),(4,2),(4,3)],5) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 5
([(0,4),(1,4),(2,3),(4,2)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 5
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 5
([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 5
([(0,3),(3,4),(4,1),(4,2)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 5
([(0,4),(2,3),(3,1),(4,2)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 5
([(0,3),(1,2),(2,4),(3,4)],5) => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 5
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
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Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see St000331The number of upper interactions of a Dyck path. and St000335The difference of lower and upper interactions. for related statistics.
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
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
The Delest-Viennot bijection $\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path $(\gamma^{(-1)}\circ\beta)(D)$.