Annals of Combinatorics最新文献

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)).

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs.

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.

We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets.

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

There has been a tremendous amount of research on the truncated theta series in the past decade. How can we understand them combinatorially? In this paper, we investigate the truncated theorems of three classical theta series of Euler and Gauss, and provide a unified combinatorial treatment. Meanwhile, we propose a possible and more direct approach to deal with these truncated theorems.

We provide a much shorter proof of Defant and Kravitz’s theorem that the length of Hitomezashi loops is congruent to 4 modulo 8. Our novel idea is to consider the length module 8 for Hitomezashi paths that take an excursion in a half-plane region.