ASMToMonotoneTriangle -- converts an ASM to a monotone triangle

Synopsis

Usage:

ASMToMonotoneTriangle A
Inputs:
- A, a matrix,
Outputs:
- a list,

Description

Converts an alternating sign matrix (ASM) to a monotone triangle according to the bijection described in [HR]. More precisely, suppose $A$ is an ASM. The unique monotone triangle $T=(T_0,\ldots,T_n)$ corresponding to $A$ has the property that $T_m$ is the (ordered) set of column indices in which $\sum_{i=1}^m A_m $ has an entry of $1$, where $A_m$ denotes the $m$th row of $A$. See [HR] for more details.

[HR]: Z. Hamaker and V. Reiner, "Weak Order and Descents for Monotone Triangles" (see arXiv:1809.10571).

i1 : A = matrix{{0,1,0,0,0,0},{0,0,0,1,0,0},{1,-1,1,-1,0,1},{0,0,0,1,0,0},{0,1,0,-1,1,0},{0,0,0,1,0,0}}

o1 = | 0 1  0 0  0 0 |
     | 0 0  0 1  0 0 |
     | 1 -1 1 -1 0 1 |
     | 0 0  0 1  0 0 |
     | 0 1  0 -1 1 0 |
     | 0 0  0 1  0 0 |

              6       6
o1 : Matrix ZZ  <-- ZZ

i2 : netList ASMToMonotoneTriangle A

     +-+-+-+-+-+-+
o2 = | | | | | | |
     +-+-+-+-+-+-+
     |2| | | | | |
     +-+-+-+-+-+-+
     |2|4| | | | |
     +-+-+-+-+-+-+
     |1|3|6| | | |
     +-+-+-+-+-+-+
     |1|3|4|6| | |
     +-+-+-+-+-+-+
     |1|2|3|5|6| |
     +-+-+-+-+-+-+
     |1|2|3|4|5|6|
     +-+-+-+-+-+-+

Ways to use ASMToMonotoneTriangle :

ASMToMonotoneTriangle(Matrix)

For the programmer

The object ASMToMonotoneTriangle is a method function.