I take it that the engine does not even TRY to start, but just spins when the starter is engaged. You need to make the test that mownie suggested. There are probably more safety interlocks than just the seat switch, and one of them could be "shorting" out the system. There is likely one on the clutch/brake pedal, and another on the PTO lever (the one that engages the blades), and one on the shift mechanism (to prevent starting if the transmission is in gear.) Testing these will require a multimeter. There is also a ground wire from the engine to the frame; typically on Husqvarna/AYP/Electrolux/Roper machines, it is a rather small black wire that is attached to the underside of the tractor frame just below the engine and inside the right front wheel. You've likely got a bad magneto or (more likely) either an electrical short caused by chafed insulation, corrosion, or loose connection, or a malfunctioning safety interlock switch. This all based on your being certain that there is no spark at the plug.

Seat switch connector has a built in "shorting bar" which will close the kill circuit when the connector is unplugged from the switch. Unplugging the connector from the switch gives the same results as if the seat is not occupied. Seat switch is just one switch in the safety interlock circuitry that can cause the engine to have no spark. The brake pedal interlock switch is actually 2 switches in one assembly. One of the brake switches is dedicated to starter inhibition, and the other switch is to kill spark and stop the engine. The PTO switch also has a circuit dedicated to the kill wire. You can disconnect the kill wire from the magneto to see if that restores spark. If doing that restores spark, something is grounding the kill wire somewhere. Could be a chafed wire.

With the new solenoid and a new key switch that pretty much leaves the PTO switch and the brake/clutch switch to test. You can test the white wires at the clutch/brake switch for the presence of 12 volts while the key is held in the start position and the PTO switch is in the engaged (blades ON) position and the clutch.brake pedal in the up position. With the PTO switch and the brake/clutch switch set as described...........only one white wire at the brake/clutch switch should be showing 12 volts (while key is in start position). If you do have 12 volts on 1 white wire at clutch/brake switch, press and hold the brake/clutch pedal down and check the other white wire for 12 volts. If no 12 volts, clutch/brake switch is defective. If you do have 12 volts on both white wires, the trouble is in the wiring that leads from the clutch/brake switch to the starter solenoid. If neither white wire at clutch/brake switch is showing 12 volts, you need to back track to the PTO switch and check the white wires on it. At the PTO switch, with the key in the start position, and PTO switch engaged as before, BOTH white wires should show 12 volts. If only 1 white wire shows 12 volts, defective PTO switch. If neither white wire on PTO switch shows 12 volts, the trouble is in the white wire between the key switch and the PTO switch.

I have a variation of the engine wont start in two parts. First I have a 917.255980 Craftsman Garden tractor, 19 hp 1993 model. Recently the primary deck belt broke and it took time to get a new one, but just today I replaced it. The battery is old, plan on replacing, but was able to get the engine started with a jump from my truck. When I flipped the switch on the deck cutters the blades started up and I was off. About 20 minutes cutting in somewhat deep grass the deck just stopped working. At first I thought it was the belt again, but when I brought the tractor back to the garage, the belt was all in place. I had turned it off to check, but when I went to start the engine nothing happened. No click, no turn over, nothing. I got my jumpers, hooked it into the truck and tried again, nothing. No click, no turnover. I am baffled why it was running and not 5 minutes later it wont do anything. Doe it have anything to do with the deck just stopping in the middle of a cut? Is the a fuse or harness connection I can giggle to get it working again?