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# monotoneTriangleToASM -- converts a monotone triangle to an ASM

## Synopsis

• Usage:
monotoneTriangleToASM M
• Inputs:
• Outputs:
• ,

## 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 .