- FindStat

Identifier

Mp00017: Binary trees —to 312-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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [6,5,1,2,4,3] => 11001

[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [3,6,1,2,5,4] => 01001

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

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

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [2,6,1,3,5,4] => 01001

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,6,2,3,5,4] => 01001

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [6,1,2,3,5,4] => 10001

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

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

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

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [4,3,1,2,6,5] => 11001

[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [2,4,1,3,6,5] => 01001

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,4,2,3,6,5] => 01001

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [4,1,2,3,6,5] => 10001

>>> Load all 1606 entries. <<<

Map

to 312-avoiding permutation

Description

Return a 312-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 minimal element of this 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.