To further expound on this topic, the company we know as Ludwig today is actually more an offshoot of WFL than the original Ludwig & Ludwig companies (although certainly related to both).
Bill Ludwig Sr. sold his drum company to Conn in the late 1920s. He continued to work for the company for a few years but eventually left. In the mid 1930s he started over but was legally prevented from using the Ludwig name (which belonged to Conn) so he used his initials WFL for the new company. This is when the Chicago plant on North Damen Ave. that figures so prominently in Ludwig lore originated as the WFL factory. WFL also developed the "classic" lug design still in use today ( although the classic Imperial lug design was developed at Ludwig & Ludwig in the 1930s under Conn's ownership ) and WFL originated the Keystone badge design. WFL was a competitor to Ludwig but grew quickly until their main rival was Chicago's other drum company, Slingerland, which was #1 in sales thanks to Gene Krupa (but WFL managed to get Buddy Rich to defect from Slingerland, telling him that he'd always be second there while he'd be their #1 guy).
In 1955 Conn decided to get out of the drum business. They also owned the Leedy line of drums which they had bought prior to Ludwig. For years Conn operated them as separate companies but in the late 1940s merged their brands into a single line that was badged "Leedy & Ludwig". They put both up for sale. Bill Ludwig Jr. wanted the family name back although Bill Sr. felt it was a waste of money. Bill Jr. worked a deal with his bitter rival Bud Slingerland where the Ludwigs would buy the Ludwig half of the business while Slingerland would buy the Leedy half.
Once WFL had completed the purchase they almost immediately changed the name of the company to The Ludwig Drum Company but they continued to sell their drums under the WFL brand for another 3 years. Their advertising said "Ludwig Drum Co. makers of WFL Drums". They also changed the badges to read "WFL" at the top but Ludwig Drum Co. on the bottom. In late 1958 they began to transition the brand name of their products over to the Ludwig name with a new badge design. The construction method of the drums themselves was not changed in any way during this time, only the name at the top of the badge was different. Some metal die-cast parts continued to wear the WFL brand for years afterwards though, until the casting dies wore out and needed to be remade.
In the case of the drum set in question I bet if you take a close look at the WFL badges they will say "Ludwig Drum Co." on the bottom. The factory certainly would not have any qualms about shipping out a set with mixed badges at the time of the transition. Also 13/16/22 is perhaps the most common combination of drum sizes in the history of drum sets so no mystery there.
Collectors relish "factory" sets but even if the drums were put together one at a time it doesn't make them any less valid as a musical instrument.
