What're they saying when we go into the sushi restaurant?

Whernever I go into an authetic sushi joint with Mrs. RickJay and The Small Girl, they say something that sounds like “Hosh ma say.”

What might that mean?

Irasshaimase!

It means, Welcome!

What’s the appropriate response to this? Nod and smile?

Arigato”, thank you.

A general greeting, such as “Konnichiwa” (“Good Day”), or “Konbonwa” (“Good Evening”) is appropriate, though “Irasshaimase” is *welcome * in the sense of “Welcome to my business” and is said when ever someone comes through the door. Unless you are greeted face to face, a verbal response is not really necessary, and a nod and a smile is fine.

“You want me to what? My goodness!”
At least that’s what I say.

“And a big howdy-doo back atchya!”