- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions

Images

[[1,2]] => [1,2] => {{1},{2}}

[[2,2]] => [1,2] => {{1},{2}}

[[1],[2]] => [2,1] => {{1,2}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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] => {{1,2}}

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

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

[[4],[5]] => [2,1] => {{1,2}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

to cycle type

Description

Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].