Munagi, Augustine O. Alternating subsets and successions. (English) Zbl 1301.05024 Ars Comb. 110, 77-86 (2013). A succession in a sequence of integers is a pair \(x,y\) of adjacent numbers satisfies \(x\equiv y \pmod 2\). Let \(h(n,k,r)\) be the number of combinations of \(k\) elements of the set \(\{1,2,\ldots ,n\}\) containing \(r\) successions. S. M. Tanny [Can. Math. Bull. 18, 769–770 (1975; Zbl 0369.05002)] showed \[ h(n,k,0)=\binom {\lfloor \frac {n+k}{2}\rfloor }{k}+\binom {\lfloor \frac {n+k-1}{2}\rfloor }{k}. \] In this paper the author recovered the above result and obtained both recursive and exact formulas for \(h(n,k,r)\). In particular, he showed \[ h(n,k,r)=\binom {k-1}{r}h(n-r,k,0). \] Moreover, the paper ends with extending the above results to the case of combinations of \(k\) elements of the set \(\{1,2,\ldots ,n\}\) whose elements arranged on a circle. Reviewer: Toufik Mansour (Haifa) Cited in 5 Documents MSC: 05A15 Exact enumeration problems, generating functions Keywords:alternating subsets; combinations; successions Citations:Zbl 0369.05002 PDFBibTeX XMLCite \textit{A. O. Munagi}, Ars Comb. 110, 77--86 (2013; Zbl 1301.05024)