On the magnitude of the roots of some well-known enumerative polynomials (Q2330074)

The Stirling numbers count the number of partitions of a set with \(n\) elements into \(k\) non-empty open subsets. Asking that blocks in the partition are ordered we get the Lah numbers and their generalization \(r\)-Lah numbers. The Whitney type variant is the \(r\)-Whitney-Lah numbers. With these numbers one can construct the enumerative polynomials called \(r\)-Dowling, \(r\)-Lah and \(r\)-Dowling-Lah polynomials. It is known that these polynomials have simple, real non-positive roots. The author gives bounds for the roots and compute their real magnitude.