- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2507 entries. <<<

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).

Map

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.