Order

, Volume 9, Issue 1, pp 31–42

# 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