Identifier
Values
[1] => [1,0] => [.,.] => [1,0] => 0
[2] => [1,0,1,0] => [.,[.,.]] => [1,1,0,0] => 0
[1,1] => [1,1,0,0] => [[.,.],.] => [1,0,1,0] => 1
[3] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => [1,1,1,0,0,0] => 0
[2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => [1,1,0,1,0,0] => 2
[4] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 3
[2,2] => [1,1,1,0,0,0] => [[.,.],[.,.]] => [1,0,1,1,0,0] => 2
[5] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 4
[3,2] => [1,0,1,1,1,0,0,0] => [.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 4
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 6
[3,3] => [1,1,1,0,1,0,0,0] => [[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 2
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 8
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 4
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[[.,.],[.,.]]]]]] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 10
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 2
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[.,[[.,[.,.]],[.,.]]]]] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 8
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 4
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [.,[.,[[.,[.,[.,.]]],[.,.]]]] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 6
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[.,[.,[.,[.,.]]]],[.,.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [.,[[.,[.,[.,[.,.]]]],[.,.]]] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 4
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[.,[[.,.],[.,.]]],[.,.]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [[.,[.,[.,[.,[.,.]]]]],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [[.,[.,[[.,.],[.,.]]]],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => 8
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [[.,[[.,.],[.,.]]],[.,[.,.]]] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => 7
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,[.,[.,.]]]] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
  • a leaf is associated with an empty Dyck path,
  • a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
Map
logarithmic height to pruning number
Description
Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].