Identifier
Values
[1,0] => [1,0] => ([],1) => ([(0,1)],2) => 1
[1,0,1,0] => [1,1,0,0] => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,0,0] => [1,0,1,0] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 1
[1,0,1,1,0,0] => [1,0,1,1,0,0] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[1,1,0,0,1,0] => [1,1,0,1,0,0] => ([(1,2)],3) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,1,0,0] => [1,1,0,0,1,0] => ([(0,1),(0,2)],3) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => ([(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] => [1,1,0,0,1,1,0,0] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(3,1)],4) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 2
[1,1,1,0,1,0,0,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(1,4),(2,3),(4,2)],5) => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 2
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 2
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 2
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 2
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => ([(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 breadth of a lattice.
The breadth of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Map
Hessenberg poset
Description
The Hessenberg poset of a Dyck path.
Let $D$ be a Dyck path of semilength $n$, regarded as a subdiagonal path from $(0,0)$ to $(n,n)$, and let $\boldsymbol{m}_i$ be the $x$-coordinate of the $i$-th up step.
Then the Hessenberg poset (or natural unit interval order) corresponding to $D$ has elements $\{1,\dots,n\}$ with $i < j$ if $j < \boldsymbol{m}_i$.
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.
Map
inverse zeta map
Description
The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.