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 $S_{\bf c}^{(r)}$ be the set of $r$-partial sequences $(a_1, \dots, a_n)$ with $0 \leq a_i \leq c_i$ for each $i$ in $\{1, \dots, n\}$, and let $S_{\bf c}^{(r)}(1)$ be the set of members of $S_{\bf c}^{(r)}$ which have $a_1 = 1$. We say that $(a_1, \dots, a_n)$ \emph{meets} $(b_1, \dots, b_m)$ if $a_i = b_i \neq 0$ for some $i$. 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 ${\bf d} = (d_1, \dots, d_m)$ with $1 \leq d_1 \leq \dots \leq d_m$. Let $A \subseteq S_{\bf c}^{(r)}$ and $B \subseteq S_{\bf d}^{(s)}$ such that $A$ and $B$ are cross-intersecting. We show that $|A||B| \leq |S_{\bf c}^{(r)}(1)||S_{\bf d}^{(s)}(1)|$ 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 $k \geq 2$ cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve

Strongly intersecting integer partitions

We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe