Here's a nice tie in.
If it's Strombergshyttan (which is also my bet) then said book explains their code:
H is "ashtray"
32 would fall in the period 1947 - 1954 (but normally the dates refer to when they are designed, not made - a good seller can be made for years afterwards).
H32 isn't in the book, H34 is - v similar shape.
Re the book - "Swedish Glass Factories Production Catalogues 1915 - 1960" by Helmut Ricke and Lars Thor.
covers almost all the production pieces for Orrefors, Orrefors-Sandvik, Kosta, Eda, Elme and Strombergshyttan for this period.
I got my copy early in collecting swedish glass, and was not immediately impressed (where's all the pretty pictures !) however it's both saved me money in the past spotting Kosta fakes, and identifying bits with illegible signatures, so for my mind it's worth the money.