To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote:

- $x_1$ as the number of office chairs
- $x_2$ as the number of dining chairs

The objective function is to maximize profit. The profit per office chair sold is $120, and the profit per dining chair sold is $180. Therefore, the objective function can be represented algebraically as:

Maximize: $120x_1 + 180x_2$

The constraints are:
1. The company does not want to invest more than $20,000 on chairs. Given that an office chair costs $200 and a dining chair costs $250, this constraint can be represented as:
\[200x_1 + 250x_2 \leq 20,000\]

2. The estimated monthly demand is at most 130 total chairs, which translates to:
\[x_1 + x_2 \leq 130\]

3. Since we cannot have negative numbers of chairs, we also have the non-negativity constraints:
\[x_1 \geq 0\]
\[x_2 \geq 0\]

Therefore, in symbolic notation, our problem can be represented as:

```json
{
    'sym_variables': [('x1', 'number of office chairs'), ('x2', 'number of dining chairs')],
    'objective_function': 'Maximize: 120*x1 + 180*x2',
    'constraints': [
        '200*x1 + 250*x2 <= 20000',
        'x1 + x2 <= 130',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

To solve this problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("chair_optimization")

# Create variables
x1 = m.addVar(name="office_chairs", vtype=GRB.INTEGER, lb=0)
x2 = m.addVar(name="dining_chairs", vtype=GRB.INTEGER, lb=0)

# Set the objective function
m.setObjective(120*x1 + 180*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(200*x1 + 250*x2 <= 20000, name="budget_constraint")
m.addConstr(x1 + x2 <= 130, name="demand_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Office chairs: {x1.x}")
    print(f"Dining chairs: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```