Identifier
Mp00001: Alternating sign matrices to semistandard tableau via monotone trianglesSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00066: Permutations inversePermutations
Mp00204: Permutations LLPS Integer partitions
Images
=>
Cc0017;cc-rep-0Cc0019;cc-rep-1Cc0002;cc-rep-4
[[1]]=>[[1]]=>[1]=>[1]=>[1] [[1,0],[0,1]]=>[[1,1],[2]]=>[3,1,2]=>[2,3,1]=>[2,1] [[0,1],[1,0]]=>[[1,2],[2]]=>[2,1,3]=>[2,1,3]=>[2,1] [[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]=>[3,1,1,1] [[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]=>[3,1,1,1] [[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]=>[3,1,1,1] [[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]=>[3,1,1,1] [[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]=>[3,1,1,1] [[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]=>[3,1,1,1] [[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]=>[3,1,1,1]
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
LLPS
Description
The Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.
An ascent in a sequence $u = (u_1, u_2, \ldots)$ is an index $i$ such that $u_i < u_{i+1}$. Let $\mathrm{asc}(u)$ denote the number of ascents of $u$, and let
$$\mathrm{asc}^{*}(u) := \begin{cases} 0 &\textrm{if u is empty}, \\ 1 + \mathrm{asc}(u) &\textrm{otherwise}.\end{cases}$$
Given a permutation $w$ in the symmetric group $\mathfrak{S}_n$, define
$A'_k := \max_{u_1, \ldots, u_k} (\mathrm{asc}^{*}(u_1) + \cdots + \mathrm{asc}^{*}(u_k))$
where the maximum is taken over disjoint subsequences ${u_i}$ of $w$.
Then $A'_1, A'_2-A'_1, A'_3-A'_2,\dots$ is a partition of $n$. Its conjugate is the Lewis-Lyu-Pylyavskyy-Sen shape of a permutation.