Yes, the property was "sold" to the partnership. You'll need to check with local regulations to see if any type of paperwork needs to be performed in order to register the property under the LLC (I imagine there would be). However, no exchange of funds needs to take place. What happens is that you can use Form 4797 on your mom's personal tax return to state that the property was disposed of on the date it was transferred to the partnership. The "sales price" is the exact amount of basis remaining in the home at the time of the transfer.
However, your mom did not receive this money. So the amount of the sale becomes part of her basis (investment) in the partnership, in addition to any additional cash or assets she may have contributed. You also will determine what cash or assets you have contributed as your basis. The percentage of income/expense allocations do not depend automatically on the ratio of contributions to capital, but rather, percentages of entitlement to income are determined by agreement between the partners. Basis determines how much each partner can receive as distributions from the partnership assets (cash or otherwise) tax-free.
Again, your mom does neet to report a disposal of the rental home as if it were a sale for the exact price of the basis value at the time of the transfer (for a zero-sum "sale"). To report this, follow the instructions in this FAQ: https://ttlc.intuit.com/replies/3388350
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