The strict upper bound of ranks of commutator subgroups of finite $p$-groups

All groups in the abstract are finite. We define rank $d(G)$ of a $p$-group $G$ as the minimal number of generators of $G$. Let $p$ be any prime number, $k_1, \dots, k_n$ – positive integers, $n \geq 2$. By $D(k_1, \dots, k_n)$ we denote the number of sequences $i_1,\dots,i_k$ in which $k \geq 2$, $i_1,\dots,i_k$ are positive integers from $[1,n]$, $i_1 > i_2$, $i_2 \leq \dots \leq i_k$ and for any $j \in [1,n]$ number $j$ may not occur in such sequences more than $(p^{k_j}-1)$ times. We prove that for any $p$-group $G$ generated by elements $a_1,\dots,a_n$ of orders $p_1^{k_1},\dots,p_n^{k_n}$ $(n \geq 2)$ the inequality $d(G') \leq D(k_1, \dots, k_n, p)$ is true and the equality in this inequality is attainable. Also, we prove that for any $p$-group $G$ generated by elements $a_1,\dots,a_n$ of orders $p_1^{k_1},\dots,p_n^{k_n}$ $(n \geq 2)$, with elementary abelian commutator subgroup $G'$ the class of nilpotency of $G'$ does not exceed $p_1^{k_1}+\dots+p_n^{k_n}-n$ and this upper bound is also attainable.