Identifier
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Mp00030:
Dyck paths
—zeta map⟶
Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤ
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
[1,0,1,1,0,1,0,0] => [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
[1,0,1,1,1,0,0,0] => [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
[1,1,0,1,1,0,0,0] => [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
[1,1,1,0,0,0,1,0] => [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
[1,1,1,0,0,1,0,0] => [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
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Description
The breadth of a lattice.
The breadth of a lattice is the least integer b such that any join x1∨x2∨⋯∨xn, with n>b, can be expressed as a join over a proper subset of {x1,x2,…,xn}.
The breadth of a lattice is the least integer b such that any join x1∨x2∨⋯∨xn, with n>b, can be expressed as a join over a proper subset of {x1,x2,…,xn}.
Map
order ideals
Description
The lattice of order ideals of a poset.
An order ideal I in a poset P is a downward closed set, i.e., a∈I and b≤a implies b∈I. This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.
An order ideal I in a poset P is a downward closed set, i.e., a∈I and b≤a implies b∈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
zeta map
Description
The zeta map on Dyck paths.
The zeta map ζ is a bijection on Dyck paths of semilength n.
It was defined in [1, Theorem 1], see also [2, Theorem 3.15] and sends the bistatistic (area, dinv) to the bistatistic (bounce, area). It is defined by sending a Dyck path D with corresponding area sequence a=(a1,…,an) to a Dyck path as follows:
The zeta map ζ is a bijection on Dyck paths of semilength n.
It was defined in [1, Theorem 1], see also [2, Theorem 3.15] and sends the bistatistic (area, dinv) to the bistatistic (bounce, area). It is defined by sending a Dyck path D with corresponding area sequence a=(a1,…,an) to a Dyck path as follows:
- First, build an intermediate Dyck path consisting of d1 north steps, followed by d1 east steps, followed by d2 north steps and d2 east steps, and so on, where di is the number of i−1's within the sequence a.
For example, given a=(0,1,2,2,2,3,1,2), we build the path
NE NNEE NNNNEEEE NE. - Next, the rectangles between two consecutive peaks are filled. Observe that such the rectangle between the kth and the (k+1)st peak must be filled by dk east steps and dk+1 north steps. In the above example, the rectangle between the second and the third peak must be filled by 2 east and 4 north steps, the 2 being the number of 1's in a, and 4 being the number of 2's. To fill such a rectangle, scan through the sequence a from left to right, and add east or north steps whenever you see a k−1 or k, respectively. So to fill the 2×4 rectangle, we look for 1's and 2's in the sequence and see 122212, so this rectangle gets filled with ENNNEN.
The complete path we obtain in thus
NENNENNNENEEENEE.
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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