- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001882: Signed permutations ⟶ ℤ

Values

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

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

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

$F_{4} = 1 + 3\ q + 4\ q^{2} + 2\ q^{3}$

Description

The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation

$\pi\in\mathfrak H_n$ , a triple

$-n \leq i < j < k\leq n$ is an occurrence of the type-B

$231$ pattern, if

$1 \leq j < k$ ,

$\pi(i) < \pi(j)$ and

$\pi(i)$ is one larger than

$\pi(k)$ , i.e.,

$\pi(i) = \pi(k) + 1$ if

$\pi(k) \neq -1$ and

$\pi(i) = 1$ otherwise.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Map

to signed permutation

Description

The signed permutation with all signs positive.

Map

rowmotion

Description

The rowmotion of a signed permutation with respect to the sorting order.
The sorting order on signed permutations (with respect to the Coxeter element

$-n, 1, 2,\dots, n-1$ ) is defined in [1].