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# ASMToMonotoneTriangle -- converts an ASM to a monotone triangle

## Synopsis

• Usage:
ASMToMonotoneTriangle A
• Inputs:
• A, ,
• Outputs:

## 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 .