- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000779: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [4,5,1,3,6,2] => [6,5,1,4,3,2] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,5,2,4,6,3] => [1,6,5,2,4,3] => 0

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

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

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

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

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

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

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

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

[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,2,5,4,6,3] => [1,2,6,5,4,3] => 0

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

[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 0

[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [6,5,4,1,2,3] => 0

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

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

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$F_{2} = 2$

$F_{3} = 3 + q$

$F_{4} = 5 + 5\ q$

$F_{5} = 6 + 17\ q + 3\ q^{2}$

$F_{6} = 8 + 46\ q + 20\ q^{2} + 2\ q^{3}$

Description

The tier of a permutation.
This is the number of elements

$i$ such that

$[i+1,k,i]$ is an occurrence of the pattern

$[2,3,1]$ . For example,

$[3,5,6,1,2,4]$ has tier

$2$ , with witnesses

$[3,5,2]$ (or

$[3,6,2]$ ) and

$[5,6,4]$ .
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].

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

Clarke-Steingrimsson-Zeng

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map

$\Phi$ in [1, sec.3]. In particular, it satisfies

$(des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi),$
where

$des$ is the number of descents, St000021The number of descents of a permutation.,
$exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
$Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
$Ebot$ is the sum of the excedance bottoms,
$Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
$Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
$Res$ is the sum of the (right) embracing numbers,
$Ine$ is the sum of the side numbers.

Map

Lehmer code rotation

Description

Sends a permutation

$\pi$ to the unique permutation

$\tau$ (of the same length) such that every entry in the Lehmer code of

$\tau$ is cyclically one larger than the Lehmer code of

$\pi$ .