I've read most of the posts here and agree with the OP. Like some others here, I worked in law enforcement for a number of years. A sheriff's department is the sole line response (responses to 9-1-1 calls) law enforcement agency for unincorporated areas of a county. This means they need 24/7 coverage 7 days a week 365 days a year. The usually operate a 9-1-1 center.
Most agencies work 8 hour shifts, others 10 and some more than that- the days off will vary based upon this factor. Using an 8 hour plan, you need 3 officers per day to cover 24 hours. Since you cannot work officers 7 days a week, you need at least two shifts or 6 officers. If you use only 2 officers per day and have them on-call overnight, this reduces the minimum needed to 4. Of course, if anyone takes a vacation or a sick day, you have no coverage. My agency used an 80% formula for coverage- you never count on more than 80% of your people being able to work. IMO, at a minimum, Absaroka County would need to have 12 patrol deputies, 1 or 2 line supervisors, 1 or 2 investigators (detectives), and then a chief deputy. This adds up to at least 15 people plus the Sheriff.
The above minimum also means you only have 1 officer for the entire county.
EDIT
There is also court security and the jail, which would require several more deputies. I know on this show they use a "Tri-county courthouse" but in reality no such facility could legally exist. There is a tri-county jail in Ohio, but incarceration and adjudication are different issues.
One other thing- contrary to what Walt has said, there is no state police in Wyoming- like most southern/western states, they use a highway patrol that deals mainly with traffic, not general calls for service. While there is a Division of Criminal Investigation, they are not a line response agency and are only involved upon request.
A show that generally went the opposite direction on staffing was In the Heat of the Night (1988). It was a small Mississippi town that seemed to have a lot of officers (and like this one a lot of murders)- yet only 1 detective. Supervisors (Sergeant, Lieutenant, Captain) routinely answered calls for service.
Ignoring politics doesn't mean politics will ignore you.
-Pericles paraphrased in <100 characters
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