- FindStat

Identifier

Mp00255: Decorated permutations —lower permutation⟶ Permutations
St000018: Permutations ⟶ ℤ

Values

[+] => [1] => 0

[-] => [1] => 0

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

[-,+] => [2,1] => 1

[+,-] => [1,2] => 0

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

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

[+,+,+] => [1,2,3] => 0

[-,+,+] => [2,3,1] => 2

[+,-,+] => [1,3,2] => 1

[+,+,-] => [1,2,3] => 0

[-,-,+] => [3,1,2] => 2

[-,+,-] => [2,1,3] => 1

[+,-,-] => [1,2,3] => 0

[-,-,-] => [1,2,3] => 0

[+,3,2] => [1,2,3] => 0

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

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

[2,1,-] => [1,2,3] => 0

[2,3,1] => [1,2,3] => 0

[3,1,2] => [1,2,3] => 0

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

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

[+,+,+,+] => [1,2,3,4] => 0

[-,+,+,+] => [2,3,4,1] => 3

[+,-,+,+] => [1,3,4,2] => 2

[+,+,-,+] => [1,2,4,3] => 1

[+,+,+,-] => [1,2,3,4] => 0

[-,-,+,+] => [3,4,1,2] => 4

[-,+,-,+] => [2,4,1,3] => 3

[-,+,+,-] => [2,3,1,4] => 2

[+,-,-,+] => [1,4,2,3] => 2

[+,-,+,-] => [1,3,2,4] => 1

[+,+,-,-] => [1,2,3,4] => 0

[-,-,-,+] => [4,1,2,3] => 3

[-,-,+,-] => [3,1,2,4] => 2

[-,+,-,-] => [2,1,3,4] => 1

[+,-,-,-] => [1,2,3,4] => 0

[-,-,-,-] => [1,2,3,4] => 0

[+,+,4,3] => [1,2,3,4] => 0

[-,+,4,3] => [2,3,1,4] => 2

[+,-,4,3] => [1,3,2,4] => 1

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

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

[-,3,2,+] => [2,4,1,3] => 3

[+,3,2,-] => [1,2,3,4] => 0

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

[+,3,4,2] => [1,2,3,4] => 0

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

[+,4,2,3] => [1,2,3,4] => 0

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

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

[-,4,+,2] => [3,2,1,4] => 3

[+,4,-,2] => [1,2,4,3] => 1

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

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

[2,1,-,+] => [1,4,2,3] => 2

[2,1,+,-] => [1,3,2,4] => 1

[2,1,-,-] => [1,2,3,4] => 0

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

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

[2,3,1,-] => [1,2,3,4] => 0

[2,3,4,1] => [1,2,3,4] => 0

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

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

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

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

[3,1,2,-] => [1,2,3,4] => 0

[3,1,4,2] => [1,2,3,4] => 0

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

[3,-,1,+] => [1,4,3,2] => 3

[3,+,1,-] => [2,1,3,4] => 1

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

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

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

[3,4,1,2] => [1,2,3,4] => 0

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

[4,1,2,3] => [1,2,3,4] => 0

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

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

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

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

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

[4,-,+,1] => [3,1,4,2] => 3

[4,+,-,1] => [2,1,4,3] => 2

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

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

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

[+,+,+,+,+] => [1,2,3,4,5] => 0

[-,+,+,+,+] => [2,3,4,5,1] => 4

[+,-,+,+,+] => [1,3,4,5,2] => 3

[+,+,-,+,+] => [1,2,4,5,3] => 2

[+,+,+,-,+] => [1,2,3,5,4] => 1

[+,+,+,+,-] => [1,2,3,4,5] => 0

[-,-,+,+,+] => [3,4,5,1,2] => 6

[-,+,-,+,+] => [2,4,5,1,3] => 5

[-,+,+,-,+] => [2,3,5,1,4] => 4

[-,+,+,+,-] => [2,3,4,1,5] => 3

[+,-,-,+,+] => [1,4,5,2,3] => 4

[+,-,+,-,+] => [1,3,5,2,4] => 3

[+,-,+,+,-] => [1,3,4,2,5] => 2

>>> Load all 2371 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

Map

lower permutation

Description

The lower 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 $u$.