Methods for electronic flatulence detection
It looks like Hydrogen is the major component: Normal Flatus. 360mL perday. How much per fart will take some closer reading. 

Here is an Arduino flamable gas detector, it probably can sense Hydrogen: 
LM393 MQ-9, say, at 10ppm. (Some shopping legwork for a Hydrogen leak detector or flammable gas sensor is in order.) So a 36mL bolus of Hydrogen (I just guessed what volumes are emitted throughout the day to make up that 360ml, and guessed 1/10 of the total) must diffuse into a volume of 3600 Liters before it is below detection level of 10ppm. Your 10ppm sensor must be within about 100 centimeters. Looks like the under-the-seat location is the right spot.