Identifier
Mp00001: to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00061: Permutations to increasing tree
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1Cc0010;cc-rep-3
[[1]]=>[[1]]=>[1]=>[.,.] [[1,0],[0,1]]=>[[1,1],[2]]=>[3,1,2]=>[[.,.],[.,.]] [[0,1],[1,0]]=>[[1,2],[2]]=>[2,1,3]=>[[.,.],[.,.]] [[1,0,0],[0,1,0],[0,0,1]]=>[[1,1,1],[2,2],[3]]=>[6,4,5,1,2,3]=>[[[.,.],[.,.]],[.,[.,.]]] [[0,1,0],[1,0,0],[0,0,1]]=>[[1,1,2],[2,2],[3]]=>[6,3,4,1,2,5]=>[[[.,.],[.,.]],[.,[.,.]]] [[1,0,0],[0,0,1],[0,1,0]]=>[[1,1,1],[2,3],[3]]=>[5,4,6,1,2,3]=>[[[.,.],[.,.]],[.,[.,.]]] [[0,1,0],[1,-1,1],[0,1,0]]=>[[1,1,2],[2,3],[3]]=>[5,3,6,1,2,4]=>[[[.,.],[.,.]],[.,[.,.]]] [[0,0,1],[1,0,0],[0,1,0]]=>[[1,1,3],[2,3],[3]]=>[4,3,5,1,2,6]=>[[[.,.],[.,.]],[.,[.,.]]] [[0,1,0],[0,0,1],[1,0,0]]=>[[1,2,2],[2,3],[3]]=>[5,2,6,1,3,4]=>[[[.,.],[.,.]],[.,[.,.]]] [[0,0,1],[0,1,0],[1,0,0]]=>[[1,2,3],[2,3],[3]]=>[4,2,5,1,3,6]=>[[[.,.],[.,.]],[.,[.,.]]]
Map
to semistandard tableau via monotone triangles
Description
The semistandard tableau corresponding the monotone triangle of an alternating sign matrix.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.
Map
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.