Cross-intersecting integer sequences

We call $(a_1,\dots ,a_n)$ an \emph{$r$-partial sequence} if exactly $r$ of its entries are positive integers and the rest are all zero. For ${\bf c} = (c_1, \dots, c_n)$with$1 \leq c_1 \leq \dots \leq c_n$,let$S_{\bf c}^{®}$bethesetof$r$-partialsequences$(a_1, \dots, a_n)$with$0 \leq a_i \leq c_i$foreach$i$in${1, \dots, n}$,andlet$S_{\bf c}^{®}(1)$ be the set of members of $S_{\mathbf{c}}^{(r)}$ which have $a_1=1$. We say that $(a_1, \dots, a_n)$\mathit{\text{meets}}$(b_1, \dots, b_m)$if$a_i = b_i \neq 0$forsome$i$. Two sets $A$ and $B$ of sequences are said to be \emph{cross-intersecting} if each sequence in $A$ meets each sequence in $B$. Let $\mathbf{d}=(d_1,\dots ,d_m)$ with $1\le d_1\le \cdots \le d_m$. Let $A\subseteq S_{\mathbf{c}}^{(r)}$ and $B \subseteq S_{\bf d}^{(s)}$suchthat$A$and$B$ are cross-intersecting. We show that $\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }\le \mathrm{\mid }{S}_{\mathbf{c}}^{(r)}(1)\mathrm{\mid }\mathrm{\mid }{S}_{\mathbf{d}}^{(s)}(1)\mathrm{\mid }$ if either $c_1 \geq 3$and$d_1 \geq 3$or${\bf c} = {\bf d}$and$r = s = n$. We also determine the cases of equality. We obtain this by proving a general cross-intersection theorem for \emph{weighted} sets. The bound generalises to one for $k\ge 2$ cross-intersecting sets.