- FindStat

Identifier

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ

Values

[1] => [[1]] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Map

Robinson-Schensted recording tableau

Description

Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.

Map

peak composition

Description

The composition corresponding to the peak 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 peak, if $i-1$ is an ascent and $i$ is a descent.
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 peak set of $T$.