Enumerating nested and consecutive partitions
The authors prove formulas for the number of partitions of an ordered set of \(n\) objects into \(p\) parts. For nested partitions \[ \text{N}(n, p)= {1\over n}\Biggl({n\atop p}\Biggr)\Biggl({n\atop p-1}\Biggr). \] For order-consecutive partitions \[ \text{OCP}(n, p)= \sum^{p-1}_{j= 0} \Biggl({n-1\atop 2p-j-2}\Biggr) \Biggl({2p-j- 2\atop j}\Biggr). \] For order-consecutive partition sequences \[ \text{OCPS}(n, p)= \sum^{p- 1}_{k= 0} (-1)^{p- 1-k} \Biggl({p-1\atop k}\Biggr) \Biggl({n+ 2k- 1\atop 2k}\Biggr). \] Properties of \(\text{N}(n, p)\) are discussed.
scientific article; zbMATH DE number 753622