- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000047: Integer compositions ⟶ ℤ

Values

[.,.] => [1] => [1] => 1

[.,[.,.]] => [2,1] => [1,1] => 1

[[.,.],.] => [1,2] => [2] => 1

[.,[.,[.,.]]] => [3,2,1] => [1,1,1] => 1

[.,[[.,.],.]] => [2,3,1] => [2,1] => 2

[[.,.],[.,.]] => [3,1,2] => [1,2] => 1

[[.,[.,.]],.] => [2,1,3] => [1,2] => 1

[[[.,.],.],.] => [1,2,3] => [3] => 1

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,1,1,1] => 1

[.,[.,[[.,.],.]]] => [3,4,2,1] => [2,1,1] => 3

[.,[[.,.],[.,.]]] => [4,2,3,1] => [1,2,1] => 2

[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,2,1] => 2

[.,[[[.,.],.],.]] => [2,3,4,1] => [3,1] => 3

[[.,.],[.,[.,.]]] => [4,3,1,2] => [1,1,2] => 1

[[.,.],[[.,.],.]] => [3,4,1,2] => [2,2] => 3

[[.,[.,.]],[.,.]] => [4,2,1,3] => [1,1,2] => 1

[[[.,.],.],[.,.]] => [4,1,2,3] => [1,3] => 1

[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,1,2] => 1

[[.,[[.,.],.]],.] => [2,3,1,4] => [2,2] => 3

[[[.,.],[.,.]],.] => [3,1,2,4] => [1,3] => 1

[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3] => 1

[[[[.,.],.],.],.] => [1,2,3,4] => [4] => 1

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,1,1,1,1] => 1

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [2,1,1,1] => 4

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [1,2,1,1] => 3

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,1,1] => 3

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,1,1] => 6

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [1,1,2,1] => 2

[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [2,2,1] => 8

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [1,1,2,1] => 2

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [1,3,1] => 3

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,1,2,1] => 2

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [2,2,1] => 8

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [1,3,1] => 3

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,3,1] => 3

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [4,1] => 4

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [1,1,1,2] => 1

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [2,1,2] => 4

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [1,2,2] => 3

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [1,2,2] => 3

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [3,2] => 6

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [1,1,1,2] => 1

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [2,1,2] => 4

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [1,1,3] => 1

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [2,3] => 4

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [1,1,1,2] => 1

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [1,2,2] => 3

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [1,1,3] => 1

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [1,1,3] => 1

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [1,4] => 1

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,1,1,2] => 1

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [2,1,2] => 4

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [1,2,2] => 3

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,2,2] => 3

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [3,2] => 6

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [1,1,3] => 1

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [2,3] => 4

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [1,1,3] => 1

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [1,4] => 1

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,1,3] => 1

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,3] => 4

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [1,4] => 1

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,4] => 1

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5] => 1

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => 1

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,1,1,1,1] => 5

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [1,2,1,1,1] => 4

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,1,1,1] => 4

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [3,1,1,1] => 10

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [1,1,2,1,1] => 3

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [2,2,1,1] => 15

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [1,1,2,1,1] => 3

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [1,3,1,1] => 6

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,1,2,1,1] => 3

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [2,2,1,1] => 15

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [1,3,1,1] => 6

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,3,1,1] => 6

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,1,1] => 10

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [1,1,1,2,1] => 2

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [2,1,2,1] => 10

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [1,2,2,1] => 8

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [1,2,2,1] => 8

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [3,2,1] => 20

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [1,1,1,2,1] => 2

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [2,1,2,1] => 10

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [1,1,3,1] => 3

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [2,3,1] => 15

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [1,1,1,2,1] => 2

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [1,2,2,1] => 8

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [1,1,3,1] => 3

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [1,1,3,1] => 3

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [1,4,1] => 4

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,1,1,2,1] => 2

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [2,1,2,1] => 10

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [1,2,2,1] => 8

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,2,2,1] => 8

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,2,1] => 20

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [1,1,3,1] => 3

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [2,3,1] => 15

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [1,1,3,1] => 3

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [1,4,1] => 4

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Description

The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.

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.

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.