Counting permutations by their alternating runs

We find a formula for the number of permutations of [ n ] that have exactly s runs up and down. The formula is at once terminating, asymptotic, and exact. The asymptotic series is valid for n → ∞ , uniformly for s ⩽ ( 1 − ϵ ) n / log n ( ϵ > 0 ).

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Bibliographic Details
Published in:	Journal of combinatorial theory. Series A Vol. 115; no. 2; pp. 213 - 225
Main Authors:	Canfield, E. Rodney, Wilf, Herbert S.
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01-02-2008
Subjects:	Alternating runs Asymptotic series Exact formula Increasing and decreasing subsequences Permutations Permutations Asymptotic series Exact formula Alternating runs Increasing and decreasing subsequences
Online Access:	Get full text
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Description
Summary:	We find a formula for the number of permutations of [ n ] that have exactly s runs up and down. The formula is at once terminating, asymptotic, and exact. The asymptotic series is valid for n → ∞ , uniformly for s ⩽ ( 1 − ϵ ) n / log n ( ϵ > 0 ).
ISSN:	0097-3165 1096-0899
DOI:	10.1016/j.jcta.2007.05.006