Related papers: Lattice points and simultaneous core partitions

Tewodros Amdeberhan and Armin Straub initiated the study of enumerating subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the…

Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing…

We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP.…

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop…

In this paper, we generalize the Catalan number to the $(n,k)$-th Catalan numbers and find a combinatorial description that the $(n,k)$-th Catalan numbers is equal to the number of partitions of $n(k-1)+2$ polygon by $(k+1)$-gon where all…

We continue our reformulation of free dendriform algebras, dealing this time with the free dendriform trialgebra generated be Y over planar rooted trees. We propose a 'deformation' of a vectorial coding used in Part I, giving a LL-lattice…

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider…

Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams defined a set $\mathsf{NC}(a,b)$ of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$.…

The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the $n^{th}$ object from…

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $m\ge2$, a set of $m+1$ partitions of a set $\Omega$, any $m$ of which are the minimal non-trivial elements of a Cartesian lattice, either form…

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a…

Catalan's conjecture claims that the Diophantine equation $x^p-y^q=1$ admits the unique solution $3^2-2^3=1$ in integers $x,y,p,q \ge 2$. The conjecture has been finally proved by P. Mih\u{a}ilescu (2002) using the theory of cyclotomic…

For $A\in\mathbb{Z}^{m\times n}$ we investigate the behaviour of the number of lattice points in $P_A(b)=\{x\in\mathbb{R}^n:Ax\leq b\}$, depending on the varying vector $b$. It is known that this number, restricted to a cone of constant…

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and…

We investigate the number $N_{d,r}(s)$ of $(s, s+r)$-core integer partitions with $d$-distinct parts. Our first main result is a proof of a recurrence relation conjectured by Sahin in 2018. We also derive generating functions, asymptotics,…

Let $\lambda$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $\lambda$ to be the difference of its leg and arm lengths. Define $h_{1,1}(\lambda)$ to be the number of cells of $\lambda$ with hook…

We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1…

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…

We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with…

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an…