Annals of Combinatorics最新文献

Two d-dimensional simplices in (mathbb {R}^d) are neighborly if its intersection is a ((d-1))-dimensional set. A family of d-dimensional simplices in (mathbb {R}^d) is called neighborly if every two simplices of the family are neighborly. Let (S_d) be the maximal cardinality of a neighborly family of d-dimensional simplices in (mathbb {R}^d). Based on the structure of some codes (Vsubset {0,1,*}^n) it is shown that (lim _{drightarrow infty }(2^{d+1}-S_d)=infty ). Moreover, a result on the structure of codes (Vsubset {0,1,*}^n) is given.

For an arbitrary set or multiset A of positive integers, we associate the A-partition function (p_A(n)) (that is the number of partitions of n whose parts belong to A). We also consider the analogue of the k-colored partition function, namely, (p_{A,-k}(n)). Further, we define a family of polynomials (f_{A,n}(x)) which satisfy the equality (f_{A,n}(k)=p_{A,-k}(n)) for all (nin mathbb {Z}_{ge 0}) and (kin mathbb {N}). This paper concerns a polynomialization of the Bessenrodt–Ono inequality, namely

where a, b are positive integers. We determine efficient criteria for the solutions of this inequality. Moreover, we also investigate a few basic properties related to both functions (f_{A,n}(x)) and (f_{A,n}'(x)).

We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination (a^3b)). Parallel to a complete classification by Cheung, Luk, and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings.

Inspired by a recent work of Kim, Kim and Lovejoy on two overpartition difference functions, we study some bipartition difference functions, four of which are related to Ramanujan’s identities recorded in his lost notebook. We show that they are always positive by elementary q-series transformations.

A classical Turán problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph H as a subgraph. It is well-known that the chromatic number of H is the graph parameter which describes the asymptotic behavior of this maximum. Here, we consider an analogous problem for oriented graphs, where compressibility plays the role of the chromatic number. Since any oriented graph having a directed cycle is not contained in any transitive tournament, it makes sense to consider only acyclic oriented graphs as forbidden subgraphs. We provide basic properties of the compressibility, show that the compressibility of acyclic oriented graphs with out-degree at most 2 is polynomial with respect to the maximum length of a directed path, and that the same holds for a larger out-degree bound if the Erdős–Hajnal conjecture is true. Additionally, generalizing previous results on powers of paths and arbitrary orientations of cycles, we determine the compressibility of acyclic oriented graphs with restricted distances of vertices to sinks and sources.

As a spin analog of the plethystic Murnaghan–Nakayama rule for Schur functions, the plethystic Murnaghan–Nakayama rule for Schur Q-functions is established with the help of the vertex operator realization. This generalizes both the Murnaghan–Nakayama rule and the Pieri rule for Schur Q-functions. A plethystic Murnaghan–Nakayama rule for Hall–Littlewood functions is also investigated.