О ранге коммутанта конечной $p$-группы, порожденной элементами порядка $p$ при $p > 2$

All groups in the abstract are finite. We define rank $d(G)$ of a $p$-group $G$ as the minimal number of generators of $G$, $d(G) = 0$ if $|G|=1$. Let $p$ be an odd prime number, $n,k$ be integers, $n \geq 1$, $k \geq 1$. By $M(n,k,p)$ we denote the number of sequences $i_1,\dots,i_k$ in which $1 \leq i_1 \leq \dots \leq i_k \leq n$, all members $i_j$ are integers and in which any integer from $[1,n]$ may be present at most $(p-1)$ times. In addition we define $M(n,k,p)=0$ if $n \leq 0$ or $k < 0$ and $M(n,0,p)=1$ if $n \geq 1$. By $C(n,k,p)$ we denote $\sum\limits_{1 \leq i_2 \leq n-1} ( M(n-i_2+1,k-2,p) -2 M(n-i_2, k-p-1, p) +M(n-i_2-1, k-2p-1,p) ) (n-i_2)$. By $D(n,p)$ we denote the following sum: $\sum\limits_{k=2}^{n(p-1)} C(n,k,p)$; $D(1,p)=0$. We prove that for any $p$-group $G$ generated by $n$ elements of order $p > 2$, $d(G') \leq D(n,p)$ and that the upper bound is attainable. As an intermediate result we prove that the class of nilpotency of such group $G$ with elementary abelian commutator subgroup does not exceed $n(p-1)$ and this upper bound is also attainable.