- FindStat

Identifier

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001255: Dyck paths ⟶ ℤ

Values

[1] => [[1]] => [1] => [1,0] => 1
[2] => [[1,2]] => [1,2] => [1,0,1,0] => 3
[1,1] => [[1],[2]] => [2,1] => [1,1,0,0] => 1
[3] => [[1,2,3]] => [1,2,3] => [1,0,1,0,1,0] => 4
[2,1] => [[1,3],[2]] => [2,1,3] => [1,1,0,0,1,0] => 3
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,1,1,0,0,0] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 5
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 4
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 5
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 3
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 6
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 5
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => 6
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 4
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 5
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 3
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 7
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0] => 7
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0] => 7
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,1,1,1,0,0,1,0,0,0,1,0] => 6
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => 7
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 5
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 3
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1

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$F_{1} = q$

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

$F_{3} = q + q^{3} + q^{4}$

$F_{4} = q + q^{3} + q^{4} + 2\ q^{5}$

$F_{5} = q + q^{3} + q^{4} + 2\ q^{5} + 2\ q^{6}$

$F_{6} = q + q^{3} + q^{4} + 2\ q^{5} + 2\ q^{6} + 4\ q^{7}$

Description

The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

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

reading tableau

Description

Return the RSK recording tableau of the reading word of the (standard) tableau

$T$ labeled down (in English convention) each column to the shape of a partition.