Processing math: 100%

Identifier
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
[[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] => [[[.,.],[.,.]],[.,[.,.]]]
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,1],[2,2,2],[3,3],[4]] => [10,8,9,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,2],[2,2,2],[3,3],[4]] => [10,8,9,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,1],[2,2,3],[3,3],[4]] => [10,7,8,5,6,9,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,2],[2,2,3],[3,3],[4]] => [10,7,8,4,5,9,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,3],[2,2,3],[3,3],[4]] => [10,6,7,4,5,8,1,2,3,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,2],[2,2,3],[3,3],[4]] => [10,7,8,3,4,9,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,3],[2,2,3],[3,3],[4]] => [10,6,7,3,4,8,1,2,5,9] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,1],[2,2,2],[3,4],[4]] => [9,8,10,5,6,7,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,2],[2,2,2],[3,4],[4]] => [9,8,10,4,5,6,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,1],[2,2,3],[3,4],[4]] => [9,7,10,5,6,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,2],[2,2,3],[3,4],[4]] => [9,7,10,4,5,8,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,3],[2,2,3],[3,4],[4]] => [9,6,10,4,5,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,2],[2,2,3],[3,4],[4]] => [9,7,10,3,4,8,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,3],[2,2,3],[3,4],[4]] => [9,6,10,3,4,7,1,2,5,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,1],[2,2,4],[3,4],[4]] => [8,7,9,5,6,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,2],[2,2,4],[3,4],[4]] => [8,7,9,4,5,10,1,2,3,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,3],[2,2,4],[3,4],[4]] => [8,6,9,4,5,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [[1,1,1,4],[2,2,4],[3,4],[4]] => [7,6,8,4,5,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,2],[2,2,4],[3,4],[4]] => [8,7,9,3,4,10,1,2,5,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,3],[2,2,4],[3,4],[4]] => [8,6,9,3,4,10,1,2,5,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => [[1,1,2,4],[2,2,4],[3,4],[4]] => [7,6,8,3,4,9,1,2,5,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,1],[2,3,3],[3,4],[4]] => [9,6,10,5,7,8,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,2],[2,3,3],[3,4],[4]] => [9,6,10,4,7,8,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,3],[2,3,3],[3,4],[4]] => [9,5,10,4,6,7,1,2,3,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,2],[2,3,3],[3,4],[4]] => [9,6,10,3,7,8,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,3],[2,3,3],[3,4],[4]] => [9,5,10,3,6,7,1,2,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,1],[2,3,4],[3,4],[4]] => [8,6,9,5,7,10,1,2,3,4] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,2],[2,3,4],[3,4],[4]] => [8,6,9,4,7,10,1,2,3,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,3],[2,3,4],[3,4],[4]] => [8,5,9,4,6,10,1,2,3,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [[1,1,1,4],[2,3,4],[3,4],[4]] => [7,5,8,4,6,9,1,2,3,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,2],[2,3,4],[3,4],[4]] => [8,6,9,3,7,10,1,2,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,3],[2,3,4],[3,4],[4]] => [8,5,9,3,6,10,1,2,4,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,4],[2,3,4],[3,4],[4]] => [7,5,8,3,6,9,1,2,4,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [[1,1,3,3],[2,3,4],[3,4],[4]] => [8,4,9,3,5,10,1,2,6,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => [[1,1,3,4],[2,3,4],[3,4],[4]] => [7,4,8,3,5,9,1,2,6,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [[1,2,2,2],[2,3,3],[3,4],[4]] => [9,6,10,2,7,8,1,3,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [[1,2,2,3],[2,3,3],[3,4],[4]] => [9,5,10,2,6,7,1,3,4,8] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => [[1,2,2,2],[2,3,4],[3,4],[4]] => [8,6,9,2,7,10,1,3,4,5] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [[1,2,2,3],[2,3,4],[3,4],[4]] => [8,5,9,2,6,10,1,3,4,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [[1,2,2,4],[2,3,4],[3,4],[4]] => [7,5,8,2,6,9,1,3,4,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => [[1,2,3,3],[2,3,4],[3,4],[4]] => [8,4,9,2,5,10,1,3,6,7] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [[1,2,3,4],[2,3,4],[3,4],[4]] => [7,4,8,2,5,9,1,3,6,10] => [[[[.,.],[.,.]],[.,[.,.]]],[.,[.,[.,.]]]]
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
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 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 σ of length n≥1 to a root node with two subtrees L and R by splitting σ at the index σ−1(1), normalizing both sides again to permutations and sending the permutations on the left and on the right of σ−1(1) to the trees L and R, respectively.