Identifier
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1Cc0019;cc-rep-2
[[1]]=>[[1]]=>[[1]]=>[1]=>[1] [[1,0],[0,1]]=>[[1,1],[2]]=>[[1,1,2]]=>[1,2,3]=>[3] [[0,1],[1,0]]=>[[1,2],[2]]=>[[1,2,2]]=>[1,2,3]=>[3] [[1,0,0],[0,1,0],[0,0,1]]=>[[1,1,1],[2,2],[3]]=>[[1,1,1,2,2],[3]]=>[6,1,2,3,4,5]=>[1,5] [[0,1,0],[1,0,0],[0,0,1]]=>[[1,1,2],[2,2],[3]]=>[[1,1,2,2,2],[3]]=>[6,1,2,3,4,5]=>[1,5] [[1,0,0],[0,0,1],[0,1,0]]=>[[1,1,1],[2,3],[3]]=>[[1,1,1,2,3],[3]]=>[5,1,2,3,4,6]=>[1,5] [[0,1,0],[1,-1,1],[0,1,0]]=>[[1,1,2],[2,3],[3]]=>[[1,1,2,2,3],[3]]=>[5,1,2,3,4,6]=>[1,5] [[0,0,1],[1,0,0],[0,1,0]]=>[[1,1,3],[2,3],[3]]=>[[1,1,2,3,3],[3]]=>[4,1,2,3,5,6]=>[1,5] [[0,1,0],[0,0,1],[1,0,0]]=>[[1,2,2],[2,3],[3]]=>[[1,2,2,2,3],[3]]=>[5,1,2,3,4,6]=>[1,5] [[0,0,1],[0,1,0],[1,0,0]]=>[[1,2,3],[2,3],[3]]=>[[1,2,2,3,3],[3]]=>[4,1,2,3,5,6]=>[1,5] [[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]]=>[[1,1,1,1,2,2,2],[3,3],[4]]=>[10,8,9,1,2,3,4,5,6,7]=>[1,2,7] [[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]]=>[[1,1,1,2,2,2,2],[3,3],[4]]=>[10,8,9,1,2,3,4,5,6,7]=>[1,2,7] [[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]]=>[[1,1,1,1,2,2,3],[3,3],[4]]=>[10,7,8,1,2,3,4,5,6,9]=>[1,2,7] [[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]]=>[[1,1,1,2,2,2,3],[3,3],[4]]=>[10,7,8,1,2,3,4,5,6,9]=>[1,2,7] [[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]]=>[[1,1,1,2,2,3,3],[3,3],[4]]=>[10,6,7,1,2,3,4,5,8,9]=>[1,2,7] [[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]]=>[[1,1,2,2,2,2,3],[3,3],[4]]=>[10,7,8,1,2,3,4,5,6,9]=>[1,2,7] [[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]]=>[[1,1,2,2,2,3,3],[3,3],[4]]=>[10,6,7,1,2,3,4,5,8,9]=>[1,2,7] [[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]]=>[[1,1,1,1,2,2,2],[3,4],[4]]=>[9,8,10,1,2,3,4,5,6,7]=>[1,2,7] [[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]]=>[[1,1,1,2,2,2,2],[3,4],[4]]=>[9,8,10,1,2,3,4,5,6,7]=>[1,2,7] [[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]]=>[[1,1,1,1,2,2,3],[3,4],[4]]=>[9,7,10,1,2,3,4,5,6,8]=>[1,2,7] [[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]]=>[[1,1,1,2,2,2,3],[3,4],[4]]=>[9,7,10,1,2,3,4,5,6,8]=>[1,2,7] [[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]]=>[[1,1,1,2,2,3],[3,3,4],[4]]=>[9,6,7,10,1,2,3,4,5,8]=>[1,3,6] [[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]]=>[[1,1,2,2,2,2,3],[3,4],[4]]=>[9,7,10,1,2,3,4,5,6,8]=>[1,2,7] [[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]]=>[[1,1,2,2,2,3],[3,3,4],[4]]=>[9,6,7,10,1,2,3,4,5,8]=>[1,3,6] [[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]]=>[[1,1,1,1,2,2,4],[3,4],[4]]=>[8,7,9,1,2,3,4,5,6,10]=>[1,2,7] [[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]]=>[[1,1,1,2,2,2,4],[3,4],[4]]=>[8,7,9,1,2,3,4,5,6,10]=>[1,2,7] [[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]]=>[[1,1,1,2,2,4,4],[3,3],[4]]=>[8,6,7,1,2,3,4,5,9,10]=>[1,2,7] [[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]]=>[[1,1,1,2,2,4,4],[3,4],[4]]=>[7,6,8,1,2,3,4,5,9,10]=>[1,2,7] [[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]]=>[[1,1,2,2,2,2,4],[3,4],[4]]=>[8,7,9,1,2,3,4,5,6,10]=>[1,2,7] [[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]]=>[[1,1,2,2,2,4,4],[3,3],[4]]=>[8,6,7,1,2,3,4,5,9,10]=>[1,2,7] [[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]]=>[[1,1,2,2,2,4,4],[3,4],[4]]=>[7,6,8,1,2,3,4,5,9,10]=>[1,2,7] [[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]]=>[[1,1,1,1,2,3,3],[3,4],[4]]=>[9,6,10,1,2,3,4,5,7,8]=>[1,2,7] [[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]]=>[[1,1,1,2,2,3,3],[3,4],[4]]=>[9,6,10,1,2,3,4,5,7,8]=>[1,2,7] [[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]]=>[[1,1,1,2,3,3,3],[3,4],[4]]=>[9,5,10,1,2,3,4,6,7,8]=>[1,2,7] [[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]]=>[[1,1,2,2,2,3,3],[3,4],[4]]=>[9,6,10,1,2,3,4,5,7,8]=>[1,2,7] [[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]]=>[[1,1,2,2,3,3,3],[3,4],[4]]=>[9,5,10,1,2,3,4,6,7,8]=>[1,2,7] [[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]]=>[[1,1,1,1,2,3,4],[3,4],[4]]=>[8,6,9,1,2,3,4,5,7,10]=>[1,2,7] [[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]]=>[[1,1,1,2,2,3,4],[3,4],[4]]=>[8,6,9,1,2,3,4,5,7,10]=>[1,2,7] [[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]]=>[[1,1,1,2,3,3,4],[3,4],[4]]=>[8,5,9,1,2,3,4,6,7,10]=>[1,2,7] [[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]]=>[[1,1,1,2,3,4,4],[3,4],[4]]=>[7,5,8,1,2,3,4,6,9,10]=>[1,2,7] [[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]]=>[[1,1,2,2,2,3,4],[3,4],[4]]=>[8,6,9,1,2,3,4,5,7,10]=>[1,2,7] [[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]]=>[[1,1,2,2,3,3,4],[3,4],[4]]=>[8,5,9,1,2,3,4,6,7,10]=>[1,2,7] [[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]]=>[[1,1,2,2,3,4,4],[3,4],[4]]=>[7,5,8,1,2,3,4,6,9,10]=>[1,2,7] [[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]]=>[[1,1,2,3,3,3,4],[3,4],[4]]=>[8,4,9,1,2,3,5,6,7,10]=>[1,2,7] [[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]]=>[[1,1,2,3,3,4,4],[3,4],[4]]=>[7,4,8,1,2,3,5,6,9,10]=>[1,2,7] [[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]]=>[[1,2,2,2,2,3,3],[3,4],[4]]=>[9,6,10,1,2,3,4,5,7,8]=>[1,2,7] [[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]]=>[[1,2,2,2,3,3,3],[3,4],[4]]=>[9,5,10,1,2,3,4,6,7,8]=>[1,2,7] [[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]]=>[[1,2,2,2,2,3,4],[3,4],[4]]=>[8,6,9,1,2,3,4,5,7,10]=>[1,2,7] [[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]]=>[[1,2,2,2,3,3,4],[3,4],[4]]=>[8,5,9,1,2,3,4,6,7,10]=>[1,2,7] [[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]]=>[[1,2,2,2,3,4,4],[3,4],[4]]=>[7,5,8,1,2,3,4,6,9,10]=>[1,2,7] [[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]]=>[[1,2,2,3,3,3,4],[3,4],[4]]=>[8,4,9,1,2,3,5,6,7,10]=>[1,2,7] [[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]]=>[[1,2,2,3,3,4,4],[3,4],[4]]=>[7,4,8,1,2,3,5,6,9,10]=>[1,2,7]
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
catabolism
Description
Remove the first row of the semistandard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater or equal to than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than or equal to $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).
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
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.