Identifier
Values
[1,0] => [[1],[]] => ([],1) => ([(0,1)],2) => 1
[1,0,1,0] => [[1,1],[]] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[1,1,0,0] => [[2],[]] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[1,0,1,0,1,0] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,1,0,0] => [[2,1],[]] => ([(0,1),(0,2)],3) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,1,0,0,1,0] => [[2,2],[1]] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[1,1,0,1,0,0] => [[3],[]] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,0,0,0] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1
[1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,1,0,0] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
[1,0,1,1,0,1,0,0] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[1,1,0,1,0,1,0,0] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,0,1,0,1,0,1,0,0] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
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 number of atoms of a lattice.
An element of a lattice is an atom if it covers the least element.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
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.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
Map
order ideals
Description
The lattice of order ideals of a poset.
An order ideal $\mathcal I$ in a poset $P$ is a downward closed set, i.e., $a \in \mathcal I$ and $b \leq a$ implies $b \in \mathcal I$. This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.