- FindStat

Identifier

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000189: Posets ⟶ ℤ

Values

[1,0] => ([],1) => 1

[1,0,1,0] => ([(0,1)],2) => 2

[1,1,0,0] => ([],2) => 2

[1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 3

[1,0,1,1,0,0] => ([(0,2),(1,2)],3) => 3

[1,1,0,0,1,0] => ([(0,1),(0,2)],3) => 3

[1,1,0,1,0,0] => ([(1,2)],3) => 3

[1,1,1,0,0,0] => ([],3) => 3

[1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 4

[1,0,1,0,1,1,0,0] => ([(0,3),(1,3),(3,2)],4) => 4

[1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4

[1,0,1,1,0,1,0,0] => ([(0,3),(1,2),(2,3)],4) => 4

[1,0,1,1,1,0,0,0] => ([(0,3),(1,3),(2,3)],4) => 4

[1,1,0,0,1,0,1,0] => ([(0,3),(3,1),(3,2)],4) => 4

[1,1,0,0,1,1,0,0] => ([(0,2),(0,3),(1,2),(1,3)],4) => 4

[1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(3,1)],4) => 4

[1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3)],4) => 4

[1,1,0,1,1,0,0,0] => ([(1,3),(2,3)],4) => 4

[1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3)],4) => 4

[1,1,1,0,0,1,0,0] => ([(1,2),(1,3)],4) => 4

[1,1,1,0,1,0,0,0] => ([(2,3)],4) => 4

[1,1,1,1,0,0,0,0] => ([],4) => 4

[1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5

[1,0,1,0,1,0,1,1,0,0] => ([(0,4),(1,4),(2,3),(4,2)],5) => 5

[1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5

[1,0,1,0,1,1,0,1,0,0] => ([(0,4),(1,2),(2,4),(4,3)],5) => 5

[1,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,4),(2,4),(4,3)],5) => 5

[1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5

[1,0,1,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 5

[1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 5

[1,0,1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 5

[1,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,3),(2,3),(3,4)],5) => 5

[1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5

[1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 5

[1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(3,4)],5) => 5

[1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 5

[1,1,0,0,1,0,1,0,1,0] => ([(0,3),(3,4),(4,1),(4,2)],5) => 5

[1,1,0,0,1,0,1,1,0,0] => ([(0,4),(1,4),(4,2),(4,3)],5) => 5

[1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 5

[1,1,0,0,1,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 5

[1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 5

[1,1,0,1,0,0,1,0,1,0] => ([(0,4),(3,2),(4,1),(4,3)],5) => 5

[1,1,0,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 5

[1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 5

[1,1,0,1,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 5

[1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 5

[1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 5

[1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(3,4)],5) => 5

[1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(2,4)],5) => 5

[1,1,0,1,1,1,0,0,0,0] => ([(1,4),(2,4),(3,4)],5) => 5

[1,1,1,0,0,0,1,0,1,0] => ([(0,4),(4,1),(4,2),(4,3)],5) => 5

[1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 5

[1,1,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(4,1),(4,2)],5) => 5

[1,1,1,0,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 5

[1,1,1,0,0,1,1,0,0,0] => ([(1,3),(1,4),(2,3),(2,4)],5) => 5

[1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(4,1)],5) => 5

[1,1,1,0,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(1,4)],5) => 5

[1,1,1,0,1,0,1,0,0,0] => ([(1,4),(2,3),(2,4)],5) => 5

[1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 5

[1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 5

[1,1,1,1,0,0,0,1,0,0] => ([(1,2),(1,3),(1,4)],5) => 5

[1,1,1,1,0,0,1,0,0,0] => ([(2,3),(2,4)],5) => 5

[1,1,1,1,0,1,0,0,0,0] => ([(3,4)],5) => 5

[1,1,1,1,1,0,0,0,0,0] => ([],5) => 5

[1,0,1,0,1,0,1,0,1,0,1,0] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 6

[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) => 6

[1,0,1,0,1,0,1,1,0,1,0,0] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 6

[1,0,1,0,1,0,1,1,1,0,0,0] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[1,0,1,0,1,1,0,1,1,0,0,0] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 6

[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) => 6

[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) => 6

[1,0,1,0,1,1,1,0,1,0,0,0] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 6

[1,0,1,0,1,1,1,1,0,0,0,0] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[1,0,1,1,0,1,1,1,0,0,0,0] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[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) => 6

[1,0,1,1,1,0,1,1,0,0,0,0] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6

>>> Load all 196 entries. <<<

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

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 5\ q^{3}$

$F_{4} = 14\ q^{4}$

$F_{5} = 42\ q^{5}$

$F_{6} = 132\ q^{6}$

Description

The number of elements in 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

$j < \boldsymbol{m}_i$ .