Identifier
-
Mp00127:
Permutations
—left-to-right-maxima to Dyck path⟶
Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ 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
[2,3,1,4] => [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
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3)],4) => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [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] => [1,1,0,0,1,1,0,0] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[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] => [1,1,1,0,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(1,4)],5) => 1
[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] => [1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],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] => [1,1,1,0,0,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
[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] => [1,1,0,1,1,0,0,1,0,0] => ([(0,4),(1,2),(1,3),(3,4)],5) => 0
[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] => [1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[2,4,5,1,3] => [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
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[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
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => ([(2,4),(3,4)],5) => 0
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[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
>>> Load all 322 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
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
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
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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