- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001207: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,5]] => [1,2] => [1,2] => [1,2] => 0

[[2,5]] => [1,2] => [1,2] => [1,2] => 0

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

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

[[5,5]] => [1,2] => [1,2] => [1,2] => 0

[[1],[5]] => [2,1] => [2,1] => [2,1] => 1

[[2],[5]] => [2,1] => [2,1] => [2,1] => 1

[[3],[5]] => [2,1] => [2,1] => [2,1] => 1

[[4],[5]] => [2,1] => [2,1] => [2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

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.

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 $w_{i+1} < v_k$.

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is

$1$
$|1|4 \to 14$
$|14|2 \to 412$
$|4|1|2|5 \to 4125$
$|4|125|3 \to 45123.$

In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).