# How to define a function that has these specific properties?

Suppose $$x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}$$. For $$x,y \in \mathbb{Z}^K_{\geq 0}$$, we write $$x \succ y$$ or $$y \prec x$$ if $$x \neq y$$ and

\begin{align*} x_{i(x,y)} > y_{i(x,y)},\quad \text{ where } \quad i(x,y) := \max\{ i: x_i \neq y_i\}. \end{align*}

That is, for any two vectors $$x$$ and $$y$$ that are not equal, we let $$i(x,y)$$ be the last position on which they differ and say that $$x \succ y$$ if the coordinate of $$x$$ at $$i(x,y)$$ is larger than the corresponding coordinate of $$y$$. We write $$x \succeq y$$ if either $$x = y$$ or $$x \succ y$$, and similarly for $$x \preceq y$$. This is a total order.

For example, if $$x = (7,2,1,0,0)$$ and $$y = (6,3,1,0,0)$$ then $$y \succ x$$ because they are equal on the last three positions and the next position that they differ is the second coordinate, since 3>2 we conclude that $$y \succ x$$. This is called "reflected lexicographic order".

Now, let $$mx(x) = max\{k: x_k > 0\}$$, we are interested in defining a function $$f: \mathbb{Z}^K_{\geq 0} \rightarrow [0,K+1)$$ that has the following properties:

• $$f(0,0,\ldots,0) = 0$$
• $$mx(x) \leq f(x) < mx(x)+1$$ (Note that when one of the coordinates of x is 1 and the rest are 0, then $$f(x)= mx(x)$$, for example let $$x = (0,1,0,0,0)$$, then $$f(x)=mx(x)=2$$)
• $$f(.)$$ is strictly increasing on $$\mathbb{Z}^K_{\geq 0}$$ wrt. the total-ordering defined above
• The effect of adding a positive value to coordinate $$k$$ should be smaller than adding the same value to coordinate $$k+1,....,K$$, having all the other values fixed, sth like convexity property but I'm not sure if the exact definition of convexity applies here. For example suppose $$K=5$$, $$f(0,3,0,0,0) - f(0,2,0,0,0) \leq f(0,0,3,0,0) - f(0,0,2,0,0)$$

I could define a function that has the first three properties, but not the fourth one: For any $$x \in \mathbb{Z}^K_{\geq 0}$$, let $$g_{k}(x) = \prod_{i=k}^{K} (1+i)^{-x_i}$$ for $$k=2,\dots,K$$ and $$g_{K+1}(x) = 1$$. \begin{align} f(x) := \sum_{k=2}^{K+1} k g_k(x) \big(1 - k^{-x_{k-1}}\big). \end{align} $$f(0,3,0,0,0) - f(0,2,0,0,0) = 2.888889 - 2.666667 = 0.222222$$ but $$f(0,0,3,0,0) - f(0,0,2,0,0) = 3.9375 - 3.75 = 0.1875$$

How to define $$f(.)$$ so that it follows all the 4 properties?

PS: This is cross-post from Math.SE (I flagged it there to be migrated to mathoverflow but no one has migrated it)

• Have you tried to figure out $K=2?$ that would be a good start. – Aaron Meyerowitz yesterday