Identifier
-
Mp00127:
Permutations
—left-to-right-maxima to Dyck path⟶
Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1] => [1,0] => [1,0] => ([],1) => 0
[1,2] => [1,0,1,0] => [1,0,1,0] => ([(0,1)],2) => 0
[2,1] => [1,1,0,0] => [1,1,0,0] => ([],2) => 0
[1,2,3] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => ([(0,2),(2,1)],3) => 0
[2,1,3] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => ([(0,1),(0,2)],3) => 0
[2,3,1] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => ([(0,2),(1,2)],3) => 0
[3,1,2] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => ([],3) => 0
[3,2,1] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => ([],3) => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => ([(0,3),(2,1),(3,2)],4) => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3)],4) => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(3,1)],4) => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => ([(0,3),(1,2),(2,3)],4) => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => ([(0,3),(3,1),(3,2)],4) => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => ([(0,3),(1,3),(3,2)],4) => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => ([(1,3),(2,3)],4) => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => ([(1,3),(2,3)],4) => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3)],4) => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3)],4) => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,3),(2,3)],4) => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => ([(0,3),(1,3),(2,3)],4) => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => ([],4) => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => ([],4) => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => ([],4) => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => ([],4) => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => ([],4) => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => ([],4) => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(3,4)],5) => 0
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(3,4)],5) => 0
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => ([],5) => 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => ([],5) => 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => ([],5) => 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => ([],5) => 0
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => ([],5) => 0
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Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
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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