In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series.