Related papers: On the orthogonal Gr\"unbaum partition problem in …

A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the…

For multipartite states we consider a notion of D-symmetry. For a system of $N$ qubits it concides with usual permutational symmetry. In case of $N$ qudits ($d\geq 3$) the D-symmetry is stronger than the permutational one. For the space of…

A strong law of large numbers for $d$-dimensional random projections of the $n$-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of $[-1,1]^n$ onto $\mathbb{R}^d$…

Consider partitions, of a cardinality $(q-1)(d+1)+1$ generic subset of euclidean $d$-space, into $q$ parts whose convex hulls have a nonempty intersection. We show that if these partitions are counted with appropriate signs $\pm 1$ then the…

In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with data at infinity. We first prove the existence and uniqueness of it. Then we show that its symmetric solutions are actually half-space solutions. Our results are…

In quantum mechanics, mutually unbiased bases (MUBs) represent orthonormal bases that are as "far apart" as possible, and their classification reveals rich underlying geometric structure. Given a complex inner product space, we construct…

For positive integers $k < n$ such that $k$ divides $n$, let $(n)^k_{\hom}$ be the set of homogeneous $k$-partitions of $\{1, \dots, n\}$, that is, the set of partitions of $\{1, \dots, n\}$ into $k$ classes of the same cardinality. In the…

Holmsen, Kyn\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset…

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…

A partition $\mathcal{P}$ of $\mathbb{R}^d$ is called a $(k,\varepsilon)$-secluded partition if, for every $\vec{p} \in \mathbb{R}^d$, the ball $\overline{B}_{\infty}(\varepsilon, \vec{p})$ intersects at most $k$ members of $\mathcal{P}$. A…

A major research area in discrete geometry is to consider the best way to partition the $d$-dimensional Euclidean space $\mathbb{R}^d$ under various quality criteria. In this paper we introduce a new type of space partitioning that is…

We study a geometry of the partition function which is defined in terms of a solution of the five-term relation. It is shown that the 3-dimensional hyperbolic structure or Euclidean AdS_3 naturally arises in the classical limit of this…

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way…

In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39]…

Partite, $3$-uniform hypergraphs are $3$-uniform hypergraphs in which each hyperedge contains exactly one point from each of the $3$ disjoint vertex classes. We consider the degree sequence problem of partite, $3$-uniform hypergraphs, that…

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove…

We address the inverse Frobenius--Perron problem: given a prescribed target distribution $\rho$, find a deterministic map $M$ such that iterations of $M$ tend to $\rho$ in distribution. We show that all solutions may be written in terms of…

The vertices of any graph with $m$ edges may be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollob\'as and Thomason conjectured that the vertices of any $r$-uniform hypergraph with $m$ edges may…

Erd\H{o}s' unit distance problem and Erd\H{o}s' distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of…

Consider a $d$-dimensional closed ball $B$ whose center coincides with that of the hypercube $[0,1]^d$. Pick the radius of $B$ in such a way that the vertices of the hypercube are outside of $B$ and the midpoints of its edges in the…