Identifier
Mp00001:
Alternating sign matrices
—to semistandard tableau via monotone triangles⟶
Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Images
[[1]] => [[1]] => [1] => [1] => [1] =>
[[1,0],[0,1]] => [[1,1],[2]] => [3,1,2] => [2,3,1] => [1,3,2] => 01
[[0,1],[1,0]] => [[1,2],[2]] => [2,1,3] => [2,1,3] => [3,1,2] => 01
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [6,4,5,1,2,3] => [4,5,6,2,3,1] => [1,3,2,6,5,4] => 01011
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => [1,6,3,2,5,4] => 01011
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [5,4,6,1,2,3] => [4,5,6,2,1,3] => [3,1,2,6,5,4] => 01011
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [5,3,6,1,2,4] => [4,5,2,6,1,3] => [3,1,6,2,5,4] => 01011
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [4,3,5,1,2,6] => [4,5,2,1,3,6] => [6,3,1,2,5,4] => 01011
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [5,2,6,1,3,4] => [4,2,5,6,1,3] => [3,1,6,5,2,4] => 01011
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [4,2,5,1,3,6] => [4,2,5,1,3,6] => [6,3,1,5,2,4] => 01011
[[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] => [7,8,9,10,4,5,6,2,3,1] => [1,3,2,6,5,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,5,6,10,2,3,1] => [1,3,2,10,6,5,4,9,8,7] => 010110111
[[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] => [7,8,9,10,4,5,2,3,6,1] => [1,6,3,2,5,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,5,10,2,3,6,1] => [1,6,3,2,10,5,4,9,8,7] => 010110111
[[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] => [7,8,9,4,5,2,3,6,10,1] => [1,10,6,3,2,5,4,9,8,7] => 010110111
[[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] => [7,8,4,5,9,10,2,3,6,1] => [1,6,3,2,10,9,5,4,8,7] => 010110111
[[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] => [7,8,4,5,9,2,3,6,10,1] => [1,10,6,3,2,9,5,4,8,7] => 010110111
[[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] => [7,8,9,10,4,5,6,2,1,3] => [3,1,2,6,5,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,5,6,10,2,1,3] => [3,1,2,10,6,5,4,9,8,7] => 010110111
[[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] => [7,8,9,10,4,5,2,6,1,3] => [3,1,6,2,5,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,5,10,2,6,1,3] => [3,1,6,2,10,5,4,9,8,7] => 010110111
[[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] => [7,8,9,4,5,2,6,10,1,3] => [3,1,10,6,2,5,4,9,8,7] => 010110111
[[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] => [7,8,4,5,9,10,2,6,1,3] => [3,1,6,2,10,9,5,4,8,7] => 010110111
[[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] => [7,8,4,5,9,2,6,10,1,3] => [3,1,10,6,2,9,5,4,8,7] => 010110111
[[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] => [7,8,9,10,4,5,2,1,3,6] => [6,3,1,2,5,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,5,10,2,1,3,6] => [6,3,1,2,10,5,4,9,8,7] => 010110111
[[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] => [7,8,9,4,5,2,10,1,3,6] => [6,3,1,10,2,5,4,9,8,7] => 010110111
[[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] => [7,8,9,4,5,2,1,3,6,10] => [10,6,3,1,2,5,4,9,8,7] => 010110111
[[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] => [7,8,4,5,9,10,2,1,3,6] => [6,3,1,2,10,9,5,4,8,7] => 010110111
[[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] => [7,8,4,5,9,2,10,1,3,6] => [6,3,1,10,2,9,5,4,8,7] => 010110111
[[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] => [7,8,4,5,9,2,1,3,6,10] => [10,6,3,1,2,9,5,4,8,7] => 010110111
[[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] => [7,8,9,10,4,2,5,6,1,3] => [3,1,6,5,2,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,10,2,5,6,1,3] => [3,1,6,5,2,10,4,9,8,7] => 010110111
[[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] => [7,8,9,4,2,5,6,10,1,3] => [3,1,10,6,5,2,4,9,8,7] => 010110111
[[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] => [7,8,4,9,10,2,5,6,1,3] => [3,1,6,5,2,10,9,4,8,7] => 010110111
[[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] => [7,8,4,9,2,5,6,10,1,3] => [3,1,10,6,5,2,9,4,8,7] => 010110111
[[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] => [7,8,9,10,4,2,5,1,3,6] => [6,3,1,5,2,4,10,9,8,7] => 010110111
[[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] => [7,8,9,4,10,2,5,1,3,6] => [6,3,1,5,2,10,4,9,8,7] => 010110111
[[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] => [7,8,9,4,2,5,10,1,3,6] => [6,3,1,10,5,2,4,9,8,7] => 010110111
[[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] => [7,8,9,4,2,5,1,3,6,10] => [10,6,3,1,5,2,4,9,8,7] => 010110111
[[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] => [7,8,4,9,10,2,5,1,3,6] => [6,3,1,5,2,10,9,4,8,7] => 010110111
[[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] => [7,8,4,9,2,5,10,1,3,6] => [6,3,1,10,5,2,9,4,8,7] => 010110111
[[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] => [7,8,4,9,2,5,1,3,6,10] => [10,6,3,1,5,2,9,4,8,7] => 010110111
[[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] => [7,8,4,2,5,9,10,1,3,6] => [6,3,1,10,9,5,2,4,8,7] => 010110111
[[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] => [7,8,4,2,5,9,1,3,6,10] => [10,6,3,1,9,5,2,4,8,7] => 010110111
[[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] => [7,4,8,9,10,2,5,6,1,3] => [3,1,6,5,2,10,9,8,4,7] => 010110111
[[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] => [7,4,8,9,2,5,6,10,1,3] => [3,1,10,6,5,2,9,8,4,7] => 010110111
[[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] => [7,4,8,9,10,2,5,1,3,6] => [6,3,1,5,2,10,9,8,4,7] => 010110111
[[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] => [7,4,8,9,2,5,10,1,3,6] => [6,3,1,10,5,2,9,8,4,7] => 010110111
[[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] => [7,4,8,9,2,5,1,3,6,10] => [10,6,3,1,5,2,9,8,4,7] => 010110111
[[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] => [7,4,8,2,5,9,10,1,3,6] => [6,3,1,10,9,5,2,8,4,7] => 010110111
[[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] => [7,4,8,2,5,9,1,3,6,10] => [10,6,3,1,9,5,2,8,4,7] => 010110111
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.
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
inverse
Description
Sends a permutation to its inverse.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
descent tops
Description
The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.
searching the database
Sorry, this map was not found in the database.