1\rangle$ could be arbitrarily large if $n$ is unbounded).

2.  For ease of discussion, we assume $f$ is *increasing* without loss of generality. In fact, for a general $f$, consider the increasing function $F:{\mathbb{N}}\rightarrow{\mathbb{N}}$, $F(n)\triangleq\max\{f(1),f(2),\cdots,f(n)\}$. Then a $f$-bounded chain is always a $F$-bounded chain since $f(n)\leq F(n)$ for all $n$. So $L(d,f)\leq L(d,F)$, and we can use $L(d,F)$ as the upper bound of the chains.

Our aim is to compute $L(d,f)$ based on the number of variables $d$ and function $f$. To this end, we particularly consider ascending chains of a special form.

\[def:f-gener\]Given a function $f:{\mathbb{N}}\rightarrow{\mathbb{N}}$, a finite sequence of monomials $m_1,m_2,\cdots,m_n\in\mathcal{M}[{\pmb{x}}]$ is called $f$-generating, if $\deg(m_i)\leq f(i)$ and $m_{i+1}\notin\langle m_1,\cdots,m_i\rangle$ for all $i\geq 1$.

Then a $f$-generating monomial sequence $m_1,m_2,\cdots,m_n$ generates a strictly ascending chain of monomial ideals $I_1\subset I_2\subset\cdots\subset I_n$ satisfying $\operatorname{gdeg}(I_i)\leq f(i)$, by defining $I_i=\langle m_1,m_2,\cdots,m_i\rangle$. Moreno-Socías proved in [@GMS92] that in order to compute $L(d,f)$, it suffices to consider the ascending chains that are generated by $f$-generating monomial sequences. That is,

\[prop:monomial\]$L(d,f)$ is exactly the greatest number of monomials of $f$-generating sequences in $\mathcal{M}[{\pmb{x}}]$.

Hence, in the rest of this section, we construct the longest chain of this form. We first do this for a 