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

Teacher cognition in language teaching: A review of research on what language teachers think, know, believe, and do

This paper reviews a selection of research from the field of foreign and second language teaching into what is referred to here as teacher cognition – what teachers think, know, and believe and the relationships of these mental constructs to what teachers do in the language teaching classroom. Within a framework suggested by more general mainstream educational research on teacher cognition, language teacher cognition is here discussed with reference to three main themes: (1) cognition and prior language learning experience, (2) cognition and teacher education, and (3) cognition and classroom practice. In addition, the findings of studies into two specific curricular areas in language teaching which have been examined by teacher cognition – grammar teaching and literacy – are discussed. This review indicates that, while the study of teacher cognition has established itself on the research agenda in the field of language teaching and provided valuable insight into the mental lives of language teachers, a clear sense of unity is lacking in the work and there are several major issues in language teaching which have yet to be explored from the perspective of teacher cognition

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