In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.
A unifying theoretical framework is presented, in which the connections among Costas sequences, circular Costas sequences, Costas polynomials, the shifting property, and Welch sequences are extended to the multidimensional context. Several conjectures on multidimensional periodic Costas arrays by J. Ortiz-Ubarri et al. are proved. Furthermore, a conjecture on Costas polynomials over extension fields presented by Muratovic-Ribic et al. is showed to be a multidimensional extension of a conjecture by Golomb and Moreno on circular Costas sequences. A weaker version of said conjecture is proved by …
A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erdős-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3…
This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit $k$ into a disjoint union of chains $\mathcal{C}_μ$ indexed by partitions of size $k$. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$. Previously, we introduced a map that bui…
On the number of Tverberg partitions in the prime power case
我们给出了一个推广的下界的Vucic和Zivaljevic的Tverberg划分的数量从素数到素数幂的情况。我们的证明受到Matousek的书《Using the Borsuk-Ulam Theorem》中证明的Z_p-index版本的启发，并使用了Volovikov引理。类似地，人们获得了一个扩展的下界的数量不同分裂的一般项链的素数功率的情况下。
Linear extensions and directed clique counts via modular partitions
Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology（系统生物学中心）；SISSA（的里雅斯特高等研究学院）；Nottingham Trent University（诺丁汉特伦特大学）
Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made possible with recourse to a new trivariate Rogers--Ramanujan type identity, which concerns a family of quadruple summations appearing as generating functions for the aforementioned overpartitions. More interestingly, the derivation of this Rogers--Ramanujan type identity is relevant to a certain well-poised basic hypergeometric series.
In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of an integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an e…
