Let \(K\) be an algebraically closed field. For a positive integer \(m\) put \(F_m:=K[t]/(t^m)\). If \(I\) is a finite partially ordered set with a unique maximal element, then the study of the representation type of the category of filtered \(I\)-chains of \(F_m\)-modules is an example of a Birkhoff type problem (this name comes from similarity to the problem studied by G. Birkhoff [Proc. Lond. Math. Soc., II. Ser. 38, 385-401 (1934; Zbl 0010.34304)]).
This problem was solved by D. Simson [J. Algebra 311, No. 1, 1-30 (2007; Zbl 1123.16010)], however a proof of an important combinatorial fact was omitted there. In the paper, the authors fill this gap.