m-1$ and $e_1=E-\sum_{i=3}^{m}a_i.$ Then $\sum_{i=1}^{m-1} e_i=E$. If $E=\sum_{i= 2}^{m} a_i$, then $e_1=a_2$ and ${\mathbf a}\prec'(e_1, \dots, e_{m-1})$. If $E\geq a_1+\sum_{i= 3}^{m} a_i$, then $e_1\geq a_1\geq a_3=e_2$. Thus, $e_1 \geq  \dots \geq e_{m-1} \geq 0$, $h=\min\{i \; : e_i<a_i\}>1$, and we have $e_i=a_{i+1}$, $h\leq i \leq m-1$. Therefore, ${\mathbf a}\prec' (e_1, \dots, e_{m-1})$.

$\Box$

Given two pairs of nonincreasing sequences of integers, $({\mathbf d}, {\mathbf a})$ and $({\mathbf c}, {\mathbf b})$, in [@DoSt13 Theorem 5.1] the authors solved the problem of obtaining necessary and sufficient conditions for the existence of a sequence ${\mathbf g}$ that is majorized (in the sense of generalized majorization) by both pairs. The conditions are very involved. In the first item of the next Lemma we solve the same problem for the 1step-generalized majorization of partitions. The characterization obtained is much more simple in this case.

\[lemmapart\] Let $S, E\geq 0$ be nonnegative integers and let ${\mathbf c}=(c_1, \dots,  c_m)$, ${\mathbf d}=(d_1, \dots,  d_m)$ be partitions of nonnegative integers such that ${\mathbf c}\neq {\mathbf d}$.

Let $\ell=\max\{i \; :\; c_i\neq d_i\}$, $f=\max\{i\in\{1, \dots, \ell\}\; : \; c_i<d_{i-1}\}$ $ (d_0=+\infty),$ and $f'=\max\{i\in\{1, \dots, \ell\}\; : \; d_i<c_{i-1}\}$ $ (c_0=+\infty).$

1.  \[itS\] There exists a partition ${\mathbf g}=(g_1, \dots,  