A short proof of the Almkvist-Meurman theorem

We give a short generating function proof of the Almkvist-Meurman theorem: For integers $h$ and $k\ne0$, define the numbers $M_n(h,k)$ by $kx(e^{hx}-1)/(e^{kx}-1)=\sum_{n=0}^\infty M_n(h,k) x^n/n!$. Equivalently, $M_n(h,k) = k^n(B_n(h/k) - B_n)$, where $B_n(u)$ is the Bernoulli polynomial. Then $M_n(h,k)$ is an integer. The proof is related to Postnikov's functional equation for the generating function for intransitive trees

The decision problem of modal product logics with a diagonal, and faulty counter machines

In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal') relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first- order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them

A Survey of Alternating Permutations

This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure

Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system $A_{n-1}$. The proof is based on an explicit formula for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases

Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees

Dukes, Selig and Steingr´ımsson were supported by grant EP/M015874/1 from The Engineering and Physical Sciences Research Council. Smith was supported by grant EP/M027147/1 from The Engineering and Physical Sciences Research Council.Peer reviewedPublisher PD