On the total positivity of restricted Stirling numbers

This note shows that the matrix whose ( n , k ) entry is the number of set partitions of { 1 , … , n } into k blocks with size at most m is never totally positive for m ≥ 3 ; thus answering a question posed in [H. Han, S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, European J. Combin. 29 (2008) 1544–1554].