- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1103 entries. <<<

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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.

Map

LLPS

Description

The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.