Identifier
Values
[1] => [1,0] => [.,.] => [1,0] => 0
[2] => [1,0,1,0] => [.,[.,.]] => [1,0,1,0] => 1
[1,1] => [1,1,0,0] => [[.,.],.] => [1,1,0,0] => 0
[3] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => 2
[2,1] => [1,0,1,1,0,0] => [.,[[.,.],.]] => [1,0,1,1,0,0] => 1
[1,1,1] => [1,1,0,1,0,0] => [[[.,.],.],.] => [1,1,1,0,0,0] => 0
[4] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 3
[3,1] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 2
[2,2] => [1,1,1,0,0,0] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 4
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => 3
[3,2] => [1,0,1,1,1,0,0,0] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 3
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [.,[.,[.,[[[.,.],.],.]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[3,3] => [1,1,1,0,1,0,0,0] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 3
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [.,[.,[[[[.,.],.],.],.]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => 1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [.,[[[[[.,.],.],.],.],.]] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[[[[[.,.],.],.],.],.],.] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 4
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => 4
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => 3
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => 2
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [.,[[.,.],[[[.,.],.],.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => 1
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[.,.],[[[[.,.],.],.],.]] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 5
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 4
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[[[.,.],.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[.,[.,.]],[[[.,.],.],.]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 3
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [.,[[[.,.],.],[[.,.],.]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[[.,.],.],[[[.,.],.],.]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 6
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[.,[.,[.,.]]],[[.,.],.]] => [1,1,0,1,0,1,0,0,1,1,0,0] => 5
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [.,[[.,[[.,.],.]],[.,.]]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => 3
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[.,[[.,.],.]],[[.,.],.]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 3
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [.,[[[[.,.],.],.],[.,.]]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 1
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[.,[.,[.,[.,.]]]],[.,.]] => [1,1,0,1,0,1,0,1,0,0,1,0] => 7
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[.,[.,[[.,.],.]]],[.,.]] => [1,1,0,1,0,1,1,0,0,0,1,0] => 5
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [.,[[[.,.],[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => [1,1,1,0,0,1,1,0,0,0,1,0] => 3
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[.,[[[.,.],.],.]],[.,.]] => [1,1,0,1,1,1,0,0,0,0,1,0] => 3
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[.,[[.,.],[.,.]]],[.,.]] => [1,1,0,1,1,0,0,1,0,0,1,0] => 5
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [[[[.,.],.],[.,.]],[.,.]] => [1,1,1,1,0,0,0,1,0,0,1,0] => 3
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => [1,1,1,0,0,1,0,0,1,0,1,0] => 4
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[.,.],[.,.]],[[.,.],.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
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Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
  • a leaf is associated to the empty Dyck Word
  • a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
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
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].