Related papers: Congruences on infinite partition and partial Brau…
We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships…
We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We…
Andrews and El Bachraoui recently studied integer partitions where the smallest part is repeated a specified number of times and any other parts are distinct. Their results included two ``surprising identities'' for which they requested…
Given a finite set of points in $\mathbb{R}^d$, Tverberg's theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph structured on the family of Tverberg partitions of a given…
An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist…
We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of both the trace and…
We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the…
In this short note, we prove several infinite family of congruences for some restricted partitions introduced by Pushpa and Vasuki (2022) (thereby, also proving a conjecture of Dasappa et. al. (2023)). We also prove some isolated…
We study the problem of partitioning a given simple polygon $P$ into a minimum number of connected polygonal pieces, each of bounded size. We describe a general technique for constructing such partitions that works for several notions of…
For an arbitrary partially ordered set $P$ its {\em dual} $P^*$ is built as the collection of all monotone mappings $P\to\2$ where $\2=\{0,1\}$ with $0<1$. The set of mappings $P^*$ is proved to be a complete lattice with respect to the…
We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid $P_n$ has degree…
It is widely believed that the parity of the partition function $p(n)$ is ``random.'' Contrary to this expectation, in this note we prove the existence of infinitely many congruence relations modulo 4 among its values. For each square-free…
We prove that the lattice Eq(X) of all equivalence relations on an infinite set X contains, as a 0,1-sublattice, the 0-coproduct of two copies of itself, thus answering a question by G.M. Bergman. Hence, by using methods initiated by de…
We provide an algorithm for calculating the unramified Brauer group of a homogeneous space $X$ of a semi-simple simply connected group $H$ with finite geometric stabiliser over any field of characteristic 0. When $k$ is a number field, we…
We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has…
We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification…
For a non-empty set $X$, the collection $Top(X)$ of all topologies on $X$ sits inside the Boolean lattice $\PP(\PP(X))$ (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space $\px$. Via this…
We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$ and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of…
Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and…
Andrews, Brietzke, R\o dseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$…