Balanced Stirling numbers

Hsu and Shiue (Adv Appl Math 20(3):366–384, 1998. https://doi.org/10.1006/aama.1998.0586) defined generalized Stirling numbers, which include as special cases a wide variety of combinatorial quantities. We prove that the two kinds of central factorial numbers are fundamentally different new special cases. Our approach also yields a previously unrecognized connection between the two kinds of central factorial numbers. In order to prove our main results, we introduce balanced Stirling numbers, which specialize the generalized Stirling numbers and can be further specialized into either kind of central factorial numbers.