Although there is pedagogical support for using computer adventure and role-playing games in order to learn a second language (L2), commercial games often lack the instructional qualities for making their language comprehensible for learners. In an interdisciplinary approach, this paper proposes a technique for adapting in-game text in order to teach L2 vocabulary, grounded in research on second language acquisition and adaptive learning systems.

Let n and k be natural numbers and let S(n, k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum n j =0 (−1) j S(n, j )is nonzero for all n > 2. We prove this conjecture for all n ≡ 2 and ≡ 2 944 838 mod 3 145 728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non-vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2-adic valuation of Stirling numbers of the second kind.

Properties of intervals in the lattice of antichains of subsets of a universe of finite size are investigated. New objects and quantities in this lattice are defined.Expressions and numerical values are deduced for the number of connected antichains and the number of fully distinguishing antichains. The latter establish a connection with Stirling numbers of the second kind. Decomposition properties of intervals in the lattice of antichains are proven. A new operator allowing partitioning the full lattice in intervals derived from lower dimensional sub-lattices is introduced. Special posets underlying an interval of antichains are defined.The poset allows the derivation of a powerful formula for the size of an interval. This formula allows computing intervals in the six dimensional space. Combinatorial coefficients allowing another decomposition of the full lattice are defined.In some specific cases, related to connected components in graphs, these coefficients can be efficiently computed. This formula allows computing the size of the lattice of order 8 efficiently. This size is the number of Dedekind of order 8, the largest one known so far.

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