Identifier
Values
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 2
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 2
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 3
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 2
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 3
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 1
[3,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 3
[2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 3
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 1
[3,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 3
[3,3,2,2,2] => [1,1,0,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 2
[3,3,3,2,2] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 2
[3,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 2
[4,4,3,3] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 1
[4,4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 2
[4,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 2
[3,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 2
[4,4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 1
[4,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 2
[4,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 1
[4,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[4,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
search for individual values
searching the database for the individual values of this statistic
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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.
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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.