Using the language of runs and patterns, a peak in a sequence of integers can be interpreted as observing a fall (or descent) immediately after a rise (or ascent). In this paper, we obtain the exact distribution of the number of peaks in a permutation by using the nonhomogeneous finite Markov chain imbedding technique and an insertion procedure. As a byproduct, we also obtain the Euler numbers, which are a sequence of the number of alternating permutations. The method is extended to obtaining the joint distribution of the number of peaks and the number of falls. Several numerical examples are given to illustrate our theoretical results.
All Science Journal Classification (ASJC) codes
- Statistics and Probability