On the Number of Peemutations within a given Distance

  • vl Scfeaefer
  • Published 2010

Abstract

where \A\ denotes the cardinality of the set A. These numbers have been calculated in [1] for k e {1,2,3} and all WGN, the set of positive integers. For k = 1, it is fairly easy to show that <p(];n-l), n e N, <p(X 0) = 1, is the sequence of Fibonacci numbers. For k = 2 and k = 3, the enumeration is based on quite involved recurrences. The corresponding sequences are listed in Sloane and Puffle [4] as series Ml600 and Ml671, respectively. The main purpose of this note is to supplement these findings by providing a closed formula for <p(k\ri) when k-\-2<m<2k + 2. Note that, for « < £ + l, one obviously has (p(k;n) = n\; thus, the cases n > 2k + 3, k > 4, remain unresolved. As a by-product, we obtain a formula for the permanent of specially patterned (0,1)-matrices. The connection to the problem above is as follows: Let n, k GM, k<n~~l, be fixed, and for i e JV, Bj = {j GZ:i-k<j <i-hk}r\N, where Z is the set of all integers. Then <p(k;n) is the same as the number of systems of distinct representatives for the set {Bl9 B2,..., BJ. Defining now for ?, j e N

Cite this paper

@inproceedings{Scfeaefer2010OnTN, title={On the Number of Peemutations within a given Distance}, author={vl Scfeaefer}, year={2010} }