A vertical counter is a counter whose bits reside in separate bytes.
Boolean logic is used to increment or decrement the counter.
Well, suppose you needed to maintain eight separate 3-bit counters. You certainly could waste one bit per counter and place each in a nibble. With the vertical counter approach, the bits are efficiently stored in memory AND all eight counters are operated on in parallel. Here eight 2-bit counters are used to count the number of samples that have occurred since the last switch transition was detected. It's a decrementing counter that terminates when zero is reached.