If you enter into a purchase agreement and place money down does the seller have the right to go back and increase the price due to a mix up on their end once this purchase is on paper contract?

They're on very shaky ground. You may have to pursue them civilly in small claims court but if both parties read and signed the contract, they should be bound to the quoted price. At the very least, a judge should find there was no meeting of the minds on the agreement and void the contract.