Identifier
Values
[.,.] => [1] => [1] => [1,0] => 0
[.,[.,.]] => [2,1] => [1,2] => [1,0,1,0] => 1
[[.,.],.] => [1,2] => [1,2] => [1,0,1,0] => 1
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => [1,0,1,1,0,0] => 2
[[.,[.,.]],[.,.]] => [4,2,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 3
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => 3
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 4
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => 4
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[[.,[.,[.,[.,[.,.]]]]],[.,.]] => [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[.,[.,[[.,.],.]]]],[.,.]] => [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[.,[[.,.],[.,.]]]],[.,.]] => [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[.,[[.,[.,.]],.]]],[.,.]] => [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[.,[[[.,.],.],.]]],[.,.]] => [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[.,.],[.,[.,.]]]],[.,.]] => [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[.,.],[[.,.],.]]],[.,.]] => [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[.,[.,.]],[.,.]]],[.,.]] => [7,5,3,2,4,1,6] => [1,6,2,4,3,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[[.,.],.],[.,.]]],[.,.]] => [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[.,[.,[.,.]]],.]],[.,.]] => [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[.,[[.,.],.]],.]],[.,.]] => [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[[.,.],[.,.]],.]],[.,.]] => [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[[.,[.,.]],.],.]],[.,.]] => [7,3,2,4,5,1,6] => [1,6,2,4,5,3,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[[[[.,.],.],.],.]],[.,.]] => [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[[.,[.,[.,[.,[.,[.,.]]]]]],.] => [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[.,[.,[[.,.],.]]]]],.] => [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[.,[[.,.],[.,.]]]]],.] => [6,4,5,3,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[.,[[.,[.,.]],.]]]],.] => [5,4,6,3,2,1,7] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[.,[[[.,.],.],.]]]],.] => [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[.,.],[.,[.,.]]]]],.] => [6,5,3,4,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[.,.],[[.,.],.]]]],.] => [5,6,3,4,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[.,[.,.]],[.,.]]]],.] => [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[[.,.],.],[.,.]]]],.] => [6,3,4,5,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[.,[.,[.,.]]],.]]],.] => [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[.,[[.,.],.]],.]]],.] => [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[[.,.],[.,.]],.]]],.] => [5,3,4,6,2,1,7] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[[.,[.,.]],.],.]]],.] => [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[.,[[[[.,.],.],.],.]]],.] => [3,4,5,6,2,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,.],[.,[.,[.,.]]]]],.] => [6,5,4,2,3,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,.],[.,[[.,.],.]]]],.] => [5,6,4,2,3,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,.],[[.,.],[.,.]]]],.] => [6,4,5,2,3,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,.],[[.,[.,.]],.]]],.] => [5,4,6,2,3,1,7] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,.],[[[.,.],.],.]]],.] => [4,5,6,2,3,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[.,.]],[.,[.,.]]]],.] => [6,5,3,2,4,1,7] => [1,7,2,4,3,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[.,.]],[[.,.],.]]],.] => [5,6,3,2,4,1,7] => [1,7,2,4,3,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,.],.],[.,[.,.]]]],.] => [6,5,2,3,4,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,.],.],[[.,.],.]]],.] => [5,6,2,3,4,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[.,[.,.]]],[.,.]]],.] => [6,4,3,2,5,1,7] => [1,7,2,5,3,4,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[[.,.],.]],[.,.]]],.] => [6,3,4,2,5,1,7] => [1,7,2,5,3,4,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,.],[.,.]],[.,.]]],.] => [6,4,2,3,5,1,7] => [1,7,2,3,5,4,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,[.,.]],.],[.,.]]],.] => [6,3,2,4,5,1,7] => [1,7,2,4,5,3,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[[.,.],.],.],[.,.]]],.] => [6,2,3,4,5,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[.,[.,[.,.]]]],.]],.] => [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[.,[[.,.],.]]],.]],.] => [4,5,3,2,6,1,7] => [1,7,2,6,3,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[[.,.],[.,.]]],.]],.] => [5,3,4,2,6,1,7] => [1,7,2,6,3,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[[.,[.,.]],.]],.]],.] => [4,3,5,2,6,1,7] => [1,7,2,6,3,5,4] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[.,[[[.,.],.],.]],.]],.] => [3,4,5,2,6,1,7] => [1,7,2,6,3,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,.],[.,[.,.]]],.]],.] => [5,4,2,3,6,1,7] => [1,7,2,3,6,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,.],[[.,.],.]],.]],.] => [4,5,2,3,6,1,7] => [1,7,2,3,6,4,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,[.,.]],[.,.]],.]],.] => [5,3,2,4,6,1,7] => [1,7,2,4,6,3,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[[.,.],.],[.,.]],.]],.] => [5,2,3,4,6,1,7] => [1,7,2,3,4,6,5] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,[.,[.,.]]],.],.]],.] => [4,3,2,5,6,1,7] => [1,7,2,5,6,3,4] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[.,[[.,.],.]],.],.]],.] => [3,4,2,5,6,1,7] => [1,7,2,5,6,3,4] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[[.,.],[.,.]],.],.]],.] => [4,2,3,5,6,1,7] => [1,7,2,3,5,6,4] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[[.,[.,.]],.],.],.]],.] => [3,2,4,5,6,1,7] => [1,7,2,4,5,6,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[[.,[[[[[.,.],.],.],.],.]],.] => [2,3,4,5,6,1,7] => [1,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
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 number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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.
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.