- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[5,4,3,1],[4,3,1]] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [2,3,3] => 6

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

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

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

[[4,4,3,2],[3,3,2]] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [3,3,2] => 6

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

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

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

[[5,4,3,2],[4,3,2]] => [4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [3,3,3] => 6

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

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

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

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

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

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

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search for individual values

searching the database for the individual values of this statistic

Description

The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.

Map

valley composition

Description

The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.

Map

inner shape

Description

The inner shape of a skew partition.

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.