Counting permutations by runs

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 − )n/ logn ( > 0). © 2007 Elsevier Inc. All rights reserved.

Counting Permutations by Their Runs Up and DownE

Counting Permutations by Alternating Descents

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers: ( 1− E1x + E3 x3 3! − E4 x4 4! + E6 x6 6! − E7 x7 7! + · · · )−1 , (∗) where ∑∞ n=0Enx n/n! ...

Counting Permutations by Their Rigid Patterns

In how many permutations does the pattern τ occur exactly m times? In most cases, the answer is unknown. When we search for rigid patterns, on the other hand, we obtain exact formulas for the solution, in all cases considered. keywords: pattern, rigid pattern, permutation, block Amy N. Myers Department of Mathematics 209 South 33rd Street Philadelphia, PA 19104 [email protected] phone: 215...

Counting 1324-Avoiding Permutations

We consider permutations that avoid the pattern 1324. By studying the generating tree for such permutations, we obtain a recurrence formula for their number. A computer program provides data for the number of 1324-avoiding permutations of length up to 20.