A unifying combinatorial approach to refined little G\"ollnitz and Capparelli's companion identities

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little G\"ollnitz identities. Noticing the connection between their results and Boulet's earlier four-parameter partition generating functions, we discover a new class of partitions, called $k$-strict partitions, to generalize their results. By applying both horizontal and vertical dissections of Ferrers' diagrams with appropriate labellings, we provide a unified combinatorial treatment of their results and shed more lights on the intriguing conditions of their companion to Capparelli's identities.Comment: This is the second revision submitted to JCTA in June, comments are welcom

Refining overpartitions by properties of non-overlined parts

We study new classes of overpartitions of numbers based on the properties of non-overlined parts. Several combinatorial identities are established by means of generating functions and bijective proofs. We show that our enumeration function satisfies a pair of infinite Ramanujantype congruences modulo 3. Lastly, by conditioning on the overlined parts of overpartitions,we give a seemingly new identity between the number of overpartitions and a certain class of ordinary partition functions. A bijective proof for this theorem also includes a partial answer to a previous request for a bijection on partitions doubly restricted by divisibility and frequency

pℓp^\ell-Torsion Points In Finite Abelian Groups And Combinatorial Identities

The main aim of this article is to compute all the moments of the number of $p^\ell$-torsion elements in some type of nite abelian groups. The averages involved in these moments are those de ned for the Cohen-Lenstra heuristics for class groups and their adaptation for Tate-Shafarevich groups. In particular, we prove that the heuristic model for Tate-Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of $p$-Selmer groups of elliptic curves. For our purpose, we are led to de ne certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context.Comment: 24 page

The Semigroups B\u3csub\u3e2\u3c/sub\u3e and B\u3csub\u3e0\u3c/sub\u3e are Inherently Nonfinitely Based, as Restriction Semigroups

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, forgetting the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable modulo monoids . These results are consequences of — and discovered as a result of — an analysis of varieties of strict restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of completely r-semisimple restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation �. For example, explicit bases of identities are found for the varieties generated by B0 and B2

Combinatorial identities for binary necklaces from exact ray-splitting trace formulae

Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (P\'olya necklaces).Comment: 15 page