- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001632: Posets ⟶ ℤ

Values

[1,2] => [.,[.,.]] => ([(0,1)],2) => 1

[2,1] => [[.,.],.] => ([(0,1)],2) => 1

[1,2,3] => [.,[.,[.,.]]] => ([(0,2),(2,1)],3) => 1

[1,3,2] => [.,[[.,.],.]] => ([(0,2),(2,1)],3) => 1

[2,1,3] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 1

[2,3,1] => [[.,[.,.]],.] => ([(0,2),(2,1)],3) => 1

[3,1,2] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 1

[3,2,1] => [[[.,.],.],.] => ([(0,2),(2,1)],3) => 1

[1,2,3,4] => [.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => 1

[1,2,4,3] => [.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => 1

[1,3,2,4] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(3,2)],4) => 1

[1,3,4,2] => [.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => 1

[1,4,2,3] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(3,2)],4) => 1

[1,4,3,2] => [.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => 1

[2,1,3,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1

[2,1,4,3] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1

[2,3,1,4] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1

[2,3,4,1] => [[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => 1

[2,4,1,3] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1

[2,4,3,1] => [[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => 1

[3,1,2,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1

[3,1,4,2] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1

[3,2,1,4] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1

[3,2,4,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(3,2)],4) => 1

[3,4,1,2] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1

[3,4,2,1] => [[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => 1

[4,1,2,3] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1

[4,1,3,2] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1

[4,2,1,3] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1

[4,2,3,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(3,2)],4) => 1

[4,3,1,2] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1

[4,3,2,1] => [[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => 1

[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1

[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1

[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1

[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1

[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0

[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0

[2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1

[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1

[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0

[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0

[3,1,5,4,2] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0

[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0

[3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1

[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1

[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2

[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1

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$F_{2} = 2\ q$

$F_{3} = 6\ q$

$F_{4} = 24\ q$

$F_{5} = 16 + 80\ q + 24\ q^{2}$

$F_{6} = 72 + 440\ q + 208\ q^{2}$

Description

The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.

Map

to poset

Description

Return the poset obtained by interpreting the tree as a Hasse diagram.