Related papers: Plane partitions II: 5 1/2 symmetry classes

We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of…

In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre Leroux himself, who talked about his joint…

The partition problem is a well-known basic NP-complete problem. We mainly consider the optimization version of it in this paper. The problem has been investigated from various perspectives for a long time and can be solved efficiently in…

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every…

In this paper, we give a purely bijective proof that two different partition classes that are both combinatorial interpretations of the partition function $p_\nu(n)$, a partition function related to the third order mock theta function…

We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such…

We consider partitions of n-dimensional boxes in R^n, n>1, into a finite number of boxes with pairwise disjoint interiors. We study sets X \subseteq (0,\infty) with the Property (W_n): for every n-dimensional box P and every partition of P,…

MacMahon enumerated the plane partitions in an $a \times b \times c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more…

We asymptotically analyse the volume-random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area…

The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…

Let $p_2(n)$ denote the number of cubic partitions. In this paper, we shall present two new congruences modulo $11$ for $p_2(n)$. We also provide an elementary alternative proof of a congruence established by Chan. Furthermore, we will…

Motivated by an error-correcting generalization of Bachet's weights problem, we define and classify relaxed complete partitions. We show that these partitions enjoy a succinct description in terms of lattice points in polyhedra, with…

In this paper we present the results from a program developed by the author that finds the unitals of the known 193 projective planes of order 25.. There are several planes for which we have not found any unital. One or more than one…

By considering the $M_2$-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences $\overline{\mbox{spt}}_2(3n)\equiv \overline{\mbox{spt}}_2(3n+1)\equiv 0\pmod{3}$. Here…

We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with…

A class of associative (super) algebras is presented, which naturally generalize both the symmetric algebra $Sym(V)$ and the wedge algebra $\wedge (V)$, where $V$ is a vector-space. These algebras are in a bijection with those subsets of…

Let $A$ be a subvariety of affine space $\mathbb{A}^n$ whose irreducible components are $d$-dimensional linear or affine subspaces of $\mathbb{A}^n$. Denote by $D(A)\subset\mathbb{N}^n$ the set of exponents of standard monomials of $A$. We…

Let $\mathcal{H}$ be a separable Hilbert space and $P$ be an idempotent on $\mathcal{H}.$ We denote by $$\Gamma_{P}=\{J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and }\hbox{ } JPJ=I-P\}$$ and $$\Delta_{P}=\{J: J=J^{\ast}=J^{-1} \hbox{ }\hbox{ and…

We answer a problem posed recently by Knuth: an n-dimensional box, with edges lying on the positive coordinate axes and generic edge lengths W_1 < W_2 < ... < W_n, is dissected into n! pieces along the planes x_i = x_j. We describe which…

We investigate the set of partial partitions of a finite set, ordered by inclusion. With this ordering the set of partial partitions can be studied as an abstract simplicial complex. We use the theory of shellable nonpure complexes to find…