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# augmentedRotheDiagram -- find the Rothe diagram and rank table for a partial ASM or permutation

- Usage:
    `augmentedRotheDiagram w`
    `augmentedRotheDiagram A`
- Inputs:
  - w, a list, or A is a Matrix
- Outputs:
  - a list,

## Description

Given a permutation in 1-line notation or a partial alternating sign matrix returns the Rothe diagram (as a list) with the rank conditions at each diagram box.

    i1 : w = {2,5,4,1,3}

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

    o1 : List

    i2 : augmentedRotheDiagram(w)

    o2 = {((1, 1), 0), ((2, 1), 0), ((2, 3), 1), ((2, 4), 1), ((3, 1), 0), ((3,
         ------------------------------------------------------------------------
         3), 1)}

    o2 : List

    i3 : A = matrix{{0,1,0},{1,-1,0},{0,0,0}}

    o3 = | 0 1  0 |
         | 1 -1 0 |
         | 0 0  0 |

                  3       3
    o3 : Matrix ZZ  <-- ZZ

    i4 : augmentedRotheDiagram(A)

    o4 = {((1, 1), 0), ((2, 2), 1), ((2, 3), 1), ((3, 2), 1), ((3, 3), 1)}

    o4 : List

## Ways to use augmentedRotheDiagram:

- `augmentedRotheDiagram(List)`
- `augmentedRotheDiagram(Matrix)`

## For the programmer

The object augmentedRotheDiagram is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixSchubert/MatrixSchubertConstructionsDOC.m2:723:0.