Identifier
Values
[1,0] => ([],1) => 0
[1,0,1,0] => ([(0,1)],2) => 1
[1,1,0,0] => ([],2) => 0
[1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 2
[1,0,1,1,0,0] => ([(0,2),(1,2)],3) => 1
[1,1,0,0,1,0] => ([(0,1),(0,2)],3) => 1
[1,1,0,1,0,0] => ([(1,2)],3) => 1
[1,1,1,0,0,0] => ([],3) => 0
[1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 3
[1,0,1,0,1,1,0,0] => ([(0,3),(1,3),(3,2)],4) => 2
[1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,0,1,1,0,1,0,0] => ([(0,3),(1,2),(2,3)],4) => 2
[1,0,1,1,1,0,0,0] => ([(0,3),(1,3),(2,3)],4) => 1
[1,1,0,0,1,0,1,0] => ([(0,3),(3,1),(3,2)],4) => 2
[1,1,0,0,1,1,0,0] => ([(0,2),(0,3),(1,2),(1,3)],4) => 1
[1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(3,1)],4) => 2
[1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3)],4) => 1
[1,1,0,1,1,0,0,0] => ([(1,3),(2,3)],4) => 1
[1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3)],4) => 1
[1,1,1,0,0,1,0,0] => ([(1,2),(1,3)],4) => 1
[1,1,1,0,1,0,0,0] => ([(2,3)],4) => 1
[1,1,1,1,0,0,0,0] => ([],4) => 0
[1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,0,1,0,1,0,1,1,0,0] => ([(0,4),(1,4),(2,3),(4,2)],5) => 3
[1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[1,0,1,0,1,1,0,1,0,0] => ([(0,4),(1,2),(2,4),(4,3)],5) => 3
[1,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[1,0,1,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 3
[1,0,1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 2
[1,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,3),(2,3),(3,4)],5) => 2
[1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2
[1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[1,1,0,0,1,0,1,0,1,0] => ([(0,3),(3,4),(4,1),(4,2)],5) => 3
[1,1,0,0,1,0,1,1,0,0] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,1,0,0,1,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1
[1,1,0,1,0,0,1,0,1,0] => ([(0,4),(3,2),(4,1),(4,3)],5) => 3
[1,1,0,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 2
[1,1,0,1,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1
[1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 2
[1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(3,4)],5) => 2
[1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => ([(1,4),(2,4),(3,4)],5) => 1
[1,1,1,0,0,0,1,0,1,0] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(4,1),(4,2)],5) => 2
[1,1,1,0,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 1
[1,1,1,0,0,1,1,0,0,0] => ([(1,3),(1,4),(2,3),(2,4)],5) => 1
[1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(4,1)],5) => 2
[1,1,1,0,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[1,1,1,0,1,0,1,0,0,0] => ([(1,4),(2,3),(2,4)],5) => 1
[1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 1
[1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 1
[1,1,1,1,0,0,0,1,0,0] => ([(1,2),(1,3),(1,4)],5) => 1
[1,1,1,1,0,0,1,0,0,0] => ([(2,3),(2,4)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => ([(3,4)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => ([],5) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 4
[1,0,1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6) => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 3
[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) => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => ([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6) => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6) => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 2
[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) => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6) => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => ([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 2
[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) => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[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) => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
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Description
The rank of the poset.
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 $j < \boldsymbol{m}_i$.
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 $j < \boldsymbol{m}_i$.
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