Identifier
-
Mp00256:
Decorated permutations
—upper permutation⟶
Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001582: Permutations ⟶ ℤ
Values
[+,+] => [1,2] => [1,2] => [2,1] => 0
[-,+] => [2,1] => [2,1] => [1,2] => 1
[+,-] => [1,2] => [1,2] => [2,1] => 0
[-,-] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 1
[+,+,+] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[-,+,+] => [2,3,1] => [2,3,1] => [1,3,2] => 1
[+,-,+] => [1,3,2] => [3,1,2] => [2,1,3] => 1
[+,+,-] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[-,-,+] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[-,+,-] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[+,-,-] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[+,3,2] => [1,3,2] => [3,1,2] => [2,1,3] => 1
[-,3,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[2,1,+] => [2,3,1] => [2,3,1] => [1,3,2] => 1
[2,1,-] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [2,3,1] => [1,3,2] => 1
[3,+,1] => [2,3,1] => [2,3,1] => [1,3,2] => 1
[3,-,1] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[+,-,+,+] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => [3,2,1,4] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,+,+] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[-,+,-,+] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => [4,1,3,2] => 1
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => [4,2,1,3] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[-,-,+,-] => [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => [3,2,1,4] => 1
[-,+,4,3] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[-,-,4,3] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[+,3,2,+] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[-,3,2,+] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => [4,2,1,3] => 1
[-,3,2,-] => [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[-,3,4,2] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[+,4,2,3] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[-,4,2,3] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[+,4,+,2] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[-,4,+,2] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[+,4,-,2] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[-,4,-,2] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[2,1,+,+] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[2,1,-,+] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,+,-] => [2,3,1,4] => [2,3,1,4] => [4,1,3,2] => 1
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[2,1,4,3] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,+] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[2,3,1,-] => [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[2,4,1,3] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[2,4,+,1] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[2,4,-,1] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[3,1,2,+] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[3,1,2,-] => [2,3,1,4] => [2,3,1,4] => [4,1,3,2] => 1
[3,1,4,2] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[3,+,1,+] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[3,-,1,+] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[3,+,1,-] => [2,3,1,4] => [2,3,1,4] => [4,1,3,2] => 1
[3,-,1,-] => [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[3,+,4,1] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[3,-,4,1] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[3,4,1,2] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[3,4,2,1] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[4,1,+,2] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[4,1,-,2] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[4,+,1,3] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[4,-,1,3] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[4,+,+,1] => [2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 1
[4,-,+,1] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[4,+,-,1] => [2,4,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[4,-,-,1] => [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[4,3,1,2] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[4,3,2,1] => [3,4,1,2] => [1,3,4,2] => [2,4,3,1] => 1
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Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Map
upper permutation
Description
The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
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