Identifier
Mp00001: Alternating sign matrices to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00066: Permutations inversePermutations
Mp00064: Permutations reversePermutations
Mp00130: Permutations descent topsBinary words
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1
[[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
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
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 $\tau(i) = \sigma(n+1-i)$.
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.