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So I assume the answer is no but want to be 100% sure.
I will already have a pulldown resistor on the inverter input, the output is connected to 555 timer trigger input (it triggers on falling edge), typically when a transistor or switch is used to pull the trigger input to ground a pullup resistor is needed.
Will the inverter hold the input high at Vcc?
If it is a push-pull output, you only need a pull resistor on an output to ensure the state when power is off.
If it is open-collector/drain output then you need a pull-up resistor since the output can only pull the line LO, not HI.
So it depends on the IC you are using.
