Class Of Partitions Research Articles

Parallel Sampling of Decomposable Graphs Using Markov Chains on Junction Trees

Bayesian inference for undirected graphical models is mostly restricted to the class of decomposable graphs, as they enjoy a rich set of properties making them amenable to high-dimensional problems. While parameter inference is straightforward in this setup, inferring the underlying graph is a challenge driven by the computational difficulty in exploring the space of decomposable graphs. This work makes two contributions to address this problem. First, we provide sufficient and necessary conditions for when multi-edge perturbations maintain decomposability of the graph. Using these, we characterize a simple class of partitions that efficiently classify all edge perturbations by whether they maintain decomposability. Second, we propose a novel parallel nonreversible Markov chain Monte Carlo sampler for distributions over junction tree representations of the graph. At every step, the parallel sampler executes simultaneously all edge perturbations within a partition. Through simulations, we demonstrate the efficiency of our new edge perturbation conditions and class of partitions. We find that our parallel sampler yields improved mixing properties in comparison to the single-move variate, and outperforms current state-of-the-art methods in terms of accuracy and computational efficiency. The implementation of our work is available in the Python package parallelDG. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Billiard partitions, Fibonacci sequences, SIP classes, and quivers

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d d -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths.

Legendre Theorems for a Class of Partitions with Initial Repetitions

AbstractPartitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition-theoretic interpretations of some Rogers–Ramanujan identities due to Lucy J. Slater.

Unifying credal partitions and fuzzy orthopartitions

This work focuses on fuzzy orthopartitions and credal partitions, which are distinct mathematical models representing partitions where the membership of elements to classes is only partially known. Firstly, we show that fuzzy orthopartitions and credal partitions are special cases of generalized fuzzy orthopartitions, which we introduce in this article as a new structure for modelling partitions with uncertainty. Next, we examine the connections between credal partitions and fuzzy orthopartitions, considering that both can be seen as types of fuzzy partitions (in particular, we deal with fuzzy probabilistic and Ruspini partitions). Moreover, we find that each generalized fuzzy orthopartition corresponds to a collection of zero, one, or infinitely many credal partitions; conversely, a credal partition maps to at most one generalized fuzzy orthopartition. Finally, we identify the class of all credal partitions that coincide with fuzzy orthopartitions.

Fully Complementary Higher Dimensional Partitions

AbstractWe introduce a symmetry class for higher dimensional partitions—fully complementary higher dimensional partitions (FCPs)—and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product, we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely quasi-transpose-complementary plane partitions, are equinumerous to symmetric plane partitions.

Some separable integer partition classes

Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus [Formula: see text]. We also extend separable integer partition classes with modulus [Formula: see text] to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.

Biases among classes of rank-crank partitions (mod 11)

In this paper, we prove inequalities for ranks, cranks, and partitions among different classes modulo 11. These were conjectured by Borozenets.

Gregarious Decompositions of Complete Equipartite Graphs and Related Structures

In a finite mathematical structure with a given partition, a substructure is said to be gregarious if either it meets each partition class or it shares at most one element with each partition class. In this paper, we considered edge decompositions of graphs and hypergraphs into gregarious subgraphs and subhypergraphs. We mostly dealt with “complete equipartite” graphs and hypergraphs, where the vertex classes have the same size and precisely those edges or hyperedges of a fixed cardinality are present that do not contain more than one element from any class. Some related graph classes generated by product operations were also considered. The generalization to hypergraphs offers a wide open area for further research.

Modulo 2 congruences for partitions with initial repetitions

Motivated by Andrews’ partitions with initial repetitions, we derive parity formulas for several functions for this class of partitions. In many cases, we present an infinite family of Ramanujan-type congruences modulo 2.

Optimal structures for focusing and energy accumulation: mathematical models and intuition

Metamaterials and composite structures are able to manipulate waves and focus fields and currents in desirable directions. Designs based on spatial and temporal variation of material properties create structures forcing fluxes into specified parts of the domain or concentrating energy into arrays of progressively sharpening pulses. The paper discusses examples of focusing structures, mathematical and intuitive considerations that influence optimal design theory. The optimality requirement introduces zones of optimal composites with variable microgeometry. The observed absence of classical solutions motivates the extension of the class of optimal partitions to composites. Such materials also provide a solution to the problem of optimal design of a thermal lens focusing thermal fluxes when the incoming fluxes are not completely known in advance. An extension of designs to dynamical materials such as space–time checkerboard composites introduces metamaterials with additional capabilities that control the accumulation of energy in the propagating waves. The discussed mathematical methods of focusing and suitable properties alternation targeted on optimality are illustrated by physical examples.

Component Behaviour and Excess of Random Bipartite Graphs Near the Critical Point

The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar phase transition as that of the binomial random graph $G(n,p)$ as $p$ passes the critical point of $\frac{1}{n}$. We study the component structure of this model near to the critical point. We show that, as with $G(n,p)$, for an appropriate range of $p$ there is a unique `giant' component and we determine asymptotically its order and excess. We also give more precise results for the distribution of the number of components of a fixed order in this range of $p$. These results rely on new bounds for the number of bipartite graphs with a fixed number of vertices and edges, which we also derive.

Affine vector space partitions

AbstractAn affine vector space partition of$${{\,\textrm{AG}\,}}(n,q)$$AG(n,q)is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

Bijections, generalizations, and other properties of sequentially congruent partitions

Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these partitions. We introduce a new partition notation specific to sequentially congruent partitions which allows us to more easily study these bijections and their compositions, and we reinterpret them in terms of Young diagram transformations. We also define a generalization of sequentially congruent partitions, and we provide several new partition bijections for these generalized sequentially congruent partitions. Finally, we investigate a question of Schneider and Schneider regarding how sequentially congruent partitions fit into Andrews’ theory of partition ideals. We prove that the maximal partition ideal of sequentially congruent partitions has infinite order and is therefore not linked, and we identify its order 1 subideals.

Chain Decompositions of q, t-Catalan Numbers: Tail Extensions and Flagpole Partitions

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 $\mathcal{C}_{\mu}$ 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 builds the tail part of each chain $\mathcal{C}_{\mu}$. Our first main contribution here is to extend this map to construct larger second-order tails for each chain. Second, we introduce new classes of partitions called flagpole partitions and generalized flagpole partitions. Third, we describe a recursive construction for building the chain $\mathcal{C}_{\mu}$ for a (generalized) flagpole partition $\mu$, assuming that the chains indexed by certain specific smaller partitions (depending on $\mu$) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.

Planarity and Genus of Sparse Random Bipartite Graphs

The genus of the binomial random graph $G(n,p)$ is well understood for a wide range of $p=p(n)$. Recently, the study of the genus of the random bipartite graph $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour to that of the binomial random graph. In this paper we show that there is a threshold for planarity of the random bipartite graph at $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$.

Distance Domination in Vertex Partitioned Graphs

We treat a variation of graph domination which involves a partition (V 1, V 2,..., Vk) of the vertex set of a graph G and domination of each partition class V i over distance d where all vertices and edges of G may be used in the domination process. Strict upper bounds and extremal graphs are presented; the results are collected in three handy tables. Further, we compare a high number of partition classes and the number of dominators needed.

Improving Characteristics of LUT-Based Sequential Blocks for Cyber-Physical Systems

A method is proposed for optimizing circuits of sequential devices which are used in cyber-physical systems (CPSs) implemented using field programmable gate arrays (FPGAs). The optimizing hardware is a very important problem connected with implementing digital parts of CPSs. In this article, we discuss a case when Mealy finite state machines (FSMs) represent behaviour of sequential devices. The proposed method is aimed at optimization of FSM circuits implemented with look-up table (LUT) elements of FPGA chip. The method aims to reduce the LUT count of Mealy FSMs with extended state codes. The method is based on finding a partition of the set of internal states by classes of compatible states. To reduce LUT count, we propose a special kind of state codes named composite state codes. The composite codes include two parts. The first part includes the binary codes of states as elements of some partition class. The second part consists of the code of corresponding partition class. Using composite state codes allows us to obtain FPGA-based FSM circuits with exactly two levels of logic. If some conditions hold, then any FSM function from the first level is implemented by a single LUT. The second level is represented as a network of multiplexers. Each multiplexer generates either an FSM output or input memory function. An example of synthesis is shown. The experiments prove that the proposed approach allows us to reduce hardware compared with two methods from Vivado, JEDI-based FSMs, and extended state assignment. Depending on the complexity of an FSM, the LUT count is reduced on average from 15.46 to 68.59 percent. The advantages of the proposed approach grow with the growth of FSM complexness. An additional positive effect of the proposed method is a decrease in the latency time.

Sparse random block matrices

The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdös–Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density.

Modular plethystic isomorphisms for two-dimensional linear groups

Let E be the natural representation of the special linear group SL2(K) over an arbitrary field K. We use the two dual constructions of the symmetric power when K has prime characteristic to construct an explicit isomorphism SymmSymℓE≅SymℓSymmE. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely SymmSymℓE≅⋀mSymℓ+m−1E. We also generalise a result first proved by King, by showing that if ∇λ is the Schur functor for the partition λ and λ∘ is the complement of λ in a rectangle with ℓ+1 rows, then ∇λSymℓE≅∇λ∘SymℓE. To illustrate that the existence of such ‘plethystic isomorphisms’ is far from obvious, we end by proving that the generalisation ∇λSymℓE≅∇λ′Symℓ+ℓ(λ′)−ℓ(λ)E of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.

Improving Characteristics of LUT-Based Three-Block Mealy FSMs’ Circuits

One of the very important problems connected with FPGA-based design is reducing the hardware amount in implemented circuits. In this paper, we discuss the implementation of Mealy finite state machines (FSMs) by circuits consisting of look-up tables (LUT). A method is proposed to reduce the LUT count of three-block circuits of Mealy FSMs. The method is based on finding a partition of set of internal states by classes of compatible states. To reduce the LUT count, we propose a special kind of state code, named complex state codes. The complex codes include two parts. The first part includes the binary codes of a state as the element of some partition class. The second part consists of the code of corresponding partition class. Using complex state codes allows obtaining FPGA-based FSM circuits with exactly four logic blocks. If some conditions hold, then any FSM function from the first and second blocks is implemented by a single LUT. The third level is represented as a network of multiplexers. These multiplexers generate either additional variable encoding collections of outputs or input memory functions. The fourth level generates FSM outputs. An example of synthesis and experimental results is shown and discussed. The experiments prove that the proposed approach allows reducing hardware compared to such methods as auto and one-hot of Vivado, JEDI. Further, the proposed approach produces circuits with fewer LUTs than for three-level Mealy FSMs based on joint use of several methods of structural decomposition. The experiments show that our approach allows reducing the LUT counts on average from 11 to 77 percent. As the complexity of an FSM increases, the gain from the application of the proposed method grows; the same is true for both the FSM performance and power consumption.