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# monotoneTriangleToASM -- converts a monotone triangle to an ASM

- Usage:
    `monotoneTriangleToASM M`
- Inputs:
  - M, a list,
- Outputs:
  - a matrix,

## Description

Converts an monotone triangle to an alternating sign matrix (ASM) according to the bijection described in [HR]. More precisely, suppose $T=(T_0,\ldots,T_n)$ is a monotone triangle. The unique ASM $A$ corresponding to $T$ is given by $A_m = \mathbb{1}_{T_m} - \mathbb{1}_{T_{m-1}}$, where $A_m$ denotes the $m$th row of $A$ and $\mathbb{1}_{T_i}$ is the is a vector of length $n$ whose entries are $1$ in the positions whose indices appear in $T_i$ and $0$ otherwise. See [HR] for more details.

This function does not check that what you've given it is actually a monotone triangle before attempting to convert to an ASM.

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

    i1 : M = {{}, {2}, {2, 4}, {1, 3, 6}, {1, 3, 4, 6}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6}}

    o1 = {{}, {2}, {2, 4}, {1, 3, 6}, {1, 3, 4, 6}, {1, 2, 3, 5, 6}, {1, 2, 3, 4,
         ------------------------------------------------------------------------
         5, 6}}

    o1 : List

    i2 : monotoneTriangleToASM M

    o2 = | 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
    o2 : Matrix ZZ  <-- ZZ

## Ways to use monotoneTriangleToASM:

- `monotoneTriangleToASM(List)`

## For the programmer

The object monotoneTriangleToASM is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixSchubert/permutationMethodsDOC.m2:640:0.