TY - JOUR

T1 - On Distribution of the Number of Peaks and the Euler Numbers of Permutations

AU - Fu, James C.

AU - Lee, Wan Chen

AU - Chang, Hsing Ming

N1 - Funding Information:
The research is supported by the Natural Science and Engineering Research Council of Canada under grant A-9216. Hsing-Ming Chang acknowledges the financial support by the Ministry of Science and Technology, R.O.C., in part through the grant 106-2118-M-006-006 and the grant 110-2118-M-006-005-MY2, which enabled his participation in this research.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023/6

Y1 - 2023/6

N2 - 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.

AB - 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.

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U2 - 10.1007/s11009-023-09987-0

DO - 10.1007/s11009-023-09987-0

M3 - Article

AN - SCOPUS:85150909392

SN - 1387-5841

VL - 25

JO - Methodology and Computing in Applied Probability

JF - Methodology and Computing in Applied Probability

IS - 2

M1 - 46

ER -