- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words

Images

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

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

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

[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 11

[.,[[.,.],.]] => [2,3,1] => [1,3,2] => 01

[[.,.],[.,.]] => [3,1,2] => [2,3,1] => 01

[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 10

[[[.,.],.],.] => [1,2,3] => [1,2,3] => 00

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 111

[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,4,3,2] => 011

[.,[[.,.],[.,.]]] => [4,2,3,1] => [2,4,3,1] => 011

[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,4,3] => 101

[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,4,3] => 001

[[.,.],[.,[.,.]]] => [4,3,1,2] => [3,4,2,1] => 011

[[.,.],[[.,.],.]] => [3,4,1,2] => [3,1,4,2] => 101

[[.,[.,.]],[.,.]] => [4,2,1,3] => [3,2,4,1] => 101

[[[.,.],.],[.,.]] => [4,1,2,3] => [2,3,4,1] => 001

[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 110

[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,2,4] => 010

[[[.,.],[.,.]],.] => [3,1,2,4] => [2,3,1,4] => 010

[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 100

[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 000

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 1111

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,5,4,3,2] => 0111

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [2,5,4,3,1] => 0111

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,1,5,4,3] => 1011

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,5,4,3] => 0011

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [3,5,4,2,1] => 0111

[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [3,1,5,4,2] => 1011

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [3,2,5,4,1] => 1011

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [2,3,5,4,1] => 0011

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [3,2,1,5,4] => 1101

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,3,2,5,4] => 0101

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [2,3,1,5,4] => 0101

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,1,3,5,4] => 1001

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,5,4] => 0001

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [4,5,3,2,1] => 0111

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [4,1,5,3,2] => 1011

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [4,2,5,3,1] => 1011

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [4,2,1,5,3] => 1101

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [1,4,2,5,3] => 0101

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [4,3,5,2,1] => 1011

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [4,3,1,5,2] => 1101

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [3,4,5,2,1] => 0011

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [3,4,1,5,2] => 0101

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [4,3,2,5,1] => 1101

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [2,4,3,5,1] => 0101

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [3,4,2,5,1] => 0101

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [3,2,4,5,1] => 1001

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [2,3,4,5,1] => 0001

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 1110

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,3,2,5] => 0110

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [2,4,3,1,5] => 0110

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [2,1,4,3,5] => 1010

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,2,4,3,5] => 0010

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [3,4,2,1,5] => 0110

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [3,1,4,2,5] => 1010

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [3,2,4,1,5] => 1010

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [2,3,4,1,5] => 0010

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 1100

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,2,4,5] => 0100

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [2,3,1,4,5] => 0100

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 1000

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0000

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 11111

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,6,5,4,3,2] => 01111

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [2,6,5,4,3,1] => 01111

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,1,6,5,4,3] => 10111

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,6,5,4,3] => 00111

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 01111

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [3,1,6,5,4,2] => 10111

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [3,2,6,5,4,1] => 10111

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [2,3,6,5,4,1] => 00111

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [3,2,1,6,5,4] => 11011

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,3,2,6,5,4] => 01011

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [2,3,1,6,5,4] => 01011

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [2,1,3,6,5,4] => 10011

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,3,6,5,4] => 00011

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [4,6,5,3,2,1] => 01111

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [4,1,6,5,3,2] => 10111

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [4,2,6,5,3,1] => 10111

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [4,2,1,6,5,3] => 11011

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [1,4,2,6,5,3] => 01011

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [4,3,6,5,2,1] => 10111

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [4,3,1,6,5,2] => 11011

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [3,4,6,5,2,1] => 00111

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [3,4,1,6,5,2] => 01011

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [4,3,2,6,5,1] => 11011

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [2,4,3,6,5,1] => 01011

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [3,4,2,6,5,1] => 01011

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [3,2,4,6,5,1] => 10011

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 00011

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [4,3,2,1,6,5] => 11101

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,4,3,2,6,5] => 01101

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [2,4,3,1,6,5] => 01101

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [2,1,4,3,6,5] => 10101

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,2,4,3,6,5] => 00101

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [3,4,2,1,6,5] => 01101

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [3,1,4,2,6,5] => 10101

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [3,2,4,1,6,5] => 10101

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [2,3,4,1,6,5] => 00101

>>> Load all 1676 entries. <<<

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

Lehmer-code to major-code bijection

Description

Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.