Furnace click on and off noise from basement to upstairs through ductwork

My new high efficiency furnace makes a sheet metalic echo-like ckick sound that I hear upstairs when the thermostat comes on and goes off.

The installer told me during installation they do not put in the fabric that separates the furnace and ductwork (what is that fabric separation called?) I didn't question it because I did not understand the use of the fabric and at that time thought they knew what they were doing. They did install some on the downdraft. The installer sidesteps my question whenever I ask what causes the noise. After giving it some thought, I am wondering if the fabric separation is what prevents noise from carrying upstairs through the ductwork. My old furnace had the fabric and I did not hear any sound coming from the basement when the thermostat clicked on and off.

They will be installing the air conditioning unit in the spring. I think there is still time to modify the installation.

Does anyone think that may be the source of the problem or have any other ideas about its source--it is very annoying!

Most furnace manufacturers specify that flexible section of the ductwork to isolate the vibrations that may originate in the furnace from being coupled to the metal ductwork. In my unprofessional opinion, it was not installed per the manufacturer's instructions, and they should resolve that at no cost to you...a pro should install per the manufacturer's instructions. Now, some things are optional or only recommended, and with that, there's some wiggle room...depends on how it is stated in the installation manual.

That being said, having it may not stop that noise from being heard upstairs. That noise could be a relay, it could be the igniter (if it is a piezoelectric rather than a glow stick igniter), or maybe even the gas valve opening. Metal to metal conducts noises better, but some will still make it upstairs. Can you isolate the noise if you are down there when it happens?