g_{m+1})$ of nonnegative integers such that $\sum_{i=1}^{m+1} g_i=S$, ${\mathbf g}\prec'{\mathbf c}$ and ${\mathbf g}\prec'{\mathbf d}$ if and only if $$\label{eqGcd}
      S\leq \sum_{i=1}^{m}\min\{c_i, d_i\}+ \max\{c_f, d_{f'}\}.$$

2.  \[itEg1\] If $f>1$ and $f'>1$, there exists a partition of nonnegative integers ${\mathbf e}=(e_1, \dots,  e_{m-1})$ such that $\sum_{i=1}^{m-1} e_i=E$, ${\mathbf c}\prec' {\mathbf e}$ and ${\mathbf d}\prec' {\mathbf e}$ if and only if $$\label{eff'm1}
    E\geq \sum_{i=1}^{m}\max\{c_i, d_i\}- \max\{c_f, d_{f'}\}.$$

3.  \[itE1\] If $f=1$ or $f'=1$, there exists a partition of nonnegative integers ${\mathbf e}=(e_1, \dots,  e_{m-1})$ such that $\sum_{i=1}^{m-1} e_i=E$, ${\mathbf c}\prec' {\mathbf e}$ and ${\mathbf d}\prec' {\mathbf e}$ if and only if $$\label{eff'=1}
    \begin{array}{l}
     E= \sum_{i=1}^{m}\max\{c_i, d_i\}- \max\{c_f, d_{f'}\},\\\mbox{or}\\
    E\geq \sum_{i=1}^{m}\max\{c_i, d_i\}- \max\{c_{f+1}, d_{f'+1}\}.\end{array}$$ Equivalently, $$E= \sum_{i=2}^{m}\max\{c_i, d_i\}\mbox{ or }
    E\geq \max\{c_1, d_1\}+\sum_{i=3}^{m}\max\{c_i, d_i\}.$$

[*Proof.*]{} Let us assume that $c_\ell>d_\ell$. If $d_\ell>c_\ell$ the proof is analogous.

We have $c_{\ell-1}\geq c_\ell>d_\ell,$ hence $f'=\ell$. Moreover, $c_i\geq c_{i+1}\geq d_i,$ $f\leq i \leq \ell-1.$ Then, $c_f \geq d_f\geq d_\ell=d_{f'}$. Hence, when $c_\ell>d_\ell$, conditions (\[eqGcd\]), (\[eff’m1\]) and (\[eff’=1\]) are respectively equivalent to $$\label{eqGcdbis}
  S\leq \sum_{i=1}^{f-1}\min\{c_i, d_i\}+ c_f+ \sum_{i=f}^{m}d_i,$$ $$\label{eff'm1bis}
E\geq \sum_{i=1}^{f-1}\max\{c_i, d_i\}+ \sum_{i=f+1}^{m}c_i,$$ $$\label{eff'=1bis}
 E= \sum_{i=2}^{m}c_i\mbox{ or }
E\geq c_1+\sum_{i=3}^{m}c_i.$$

Moreover, if $f'=1$ then $\ell=1$ and as a consequence, $f=1$. Therefore, when $c_\ell>d_\ell$, the condition $f=1$ or $f'=1$ 