# The number of strictly increasing mappings of fences

Article

- Received:
- Accepted:

DOI: 10.1007/BF00419037

- Cite this article as:
- Rutkowski, A. Order (1992) 9: 31. doi:10.1007/BF00419037

## 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*a*_{n,m}+*b*_{n,m}+*c*_{n}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 06A07secondary 05A15### Key words

Fencesimilar fencesstrictly increasing mapping## Copyright information

© Kluwer Academic Publishers 1992