Order

, Volume 9, Issue 1, pp 31–42 | Cite as

The number of strictly increasing mappings of fences

  • Aleksander Rutkowski
Article

Abstract

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rfloor\). If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n−m) then there are an,m+bn,m+cn of such mappings, where
$$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$

Mathematics Subject Classifications (1991)

Primary 06A07 secondary 05A15 

Key words

Fence similar fences strictly increasing mapping 

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References

  1. 1.
    J. Riordan (1968) Combinatorial Identities, John Wiley & Sons Inc., New York-London-Sydney.Google Scholar
  2. 2.
    A. Rutkowski (1993) On strictly increasing selfmappings of fences. How many of them are there? — to appear.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Aleksander Rutkowski
    • 1
  1. 1.Institute of MathematicsWarsaw University of TechnologyWarsawPoland

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