Consider the following definition in the second page of [this article](https://pdfs.semanticscholar.org/da0b/1a3430e4be45119921b8b8a6578b355bb2f6.pdf):



> 
> For any two integers $k,n\ge 1$, there is a unique way of writing
>  $$n=\binom{a\_k}{k}+\binom{a\_{k-1}}{k-1}+\dots+\binom{a\_i}{i}$$
>  so that $a\_k > a\_{k-1} > \dots > a\_i\geq i > 0$. Define
>  $$\partial\_{k-1}(n) = \binom{a\_k}{k-1}+\binom{a\_{k-1}}{k-2}+\dots+\binom{a\_i}{i-1}.$$
> 
> 
> 


Is it true that when $k$ is fixed, the function $\partial\_{k-1}(n)$ is weakly increasing in $n$? It does not look straightforward to prove this directly, but I assume it has already been shown somewhere.



Monotonicity of $M$-sequence
The reason is that this ordering is lexicographic. We may induct on $k$. The base $k=1$ is clear. Assume that $n>m$ and
\begin{align\*}
n=\binom{a\_k}{k}+\binom{a\_{k-1}}{k-1}+\ldots+\binom{a\_i}{i},a\_k>a\_{k-1}>\ldots>a\_i\geqslant i,\\
m=\binom{b\_k}{k}+\binom{b\_{k-1}}{k-1}+\ldots+\binom{b\_j}{j},b\_k>b\_{k-1}>\ldots>b\_j\geqslant j.
\end{align\*}
If $a\_k>b\_k$, then 
$$
\partial\_{k-1}(n)\geqslant \binom{a\_k}{k-1}\geqslant 
\binom{b\_k+1}{k-1}=\binom{b\_k}{k-1}+\binom{b\_k}{k-2}=\\
\binom{b\_k}{k-1}+\binom{b\_k-1}{k-2}+\binom{b\_k-1}{k-3}=\ldots=\\
\binom{b\_k}{k-1}+\binom{b\_k-1}{k-2}+\binom{b\_k-2}{k-3}+\ldots+\binom{b\_k-k+j+1}{j}+\binom{b\_k-k+j+1}{j-1}\geqslant \\
\binom{b\_k}{k-1}+\binom{b\_{k-1}}{k-2}+\binom{b\_{k-2}}{k-3}+\ldots+\binom{b\_{j+1}}{j}+\binom{b\_{j+1}}{j-1}>\partial\_{k-1}(m).
$$
If $a\_k=b\_k$, they cancel and we use induction. If $a\_k<b\_k$, then $m>n$ by the same reasoning as above.


