Class Of Partitions Research Articles

Counting set partitions by the number of movable letters

A movable letter within a sequence belonging to a class is one that may be transposed with its predecessor while staying within the class. We consider in this paper the problem of counting finite set partitions by the number of movable letters in their canonical sequential representations. A further restricted count on the set of partitions of with k blocks is given wherein it is required that no two equal letters be transposed. Explicit formulas for the associated exponential generating functions and for the totals of the respective statistics over all members of are found. To establish several of our results, we solve explicitly various linear partial differential equations. Finally, some comparable results are found for the class of non-crossing partitions of where in this case we focus instead on the ordinary generating functions of the associated distributions.

Formalization of scientific and technical sentences in Russian language

The article presents the research results on the computational theory of semantic interpretation in Russian language. These interpretations represent a generalizing formalization of a scientific and technical style sentences. Furthermore, the concept of a partition system of a word scale, its classes and a method of forming partition classes is defined. We showed that the word scale corresponds to many partition systems that are specified by a special rule for selecting instances from classes. Additionally, we proposed generalized algorithms for constructing a partition system.

On Unbalanced Boolean Functions with Best Correlation Immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. 
 We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.

From Dyck Paths to Standard Young Tableaux

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength $n$ that is shown to be in bijection with the set of all SYT with $n$ boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions.

Fractal form of the partition functions <i>p</i> (<i>n</i>)

The fractal family $\left\{p\left(n, k\right), k \in \mathbb{N}\right\}$, describe a rule to calculate the number of partitions obtained by decomposing $n\in \mathbb{N}$, into exactly $k$ parts. In this paper, we will present a novel method for proving that polynomials $\left\{p\left(n, k\right), k \in \mathbb{N} \right\}$ have fractal form. For each class $k$, up to the $LCM\left(1, 2, 3, \dots, k\right)$, different polynomials of degree $k-1$ are needed to form one quasi-polynomial $p\left(n, k\right)$. All the polynomials (needed for the same class $k$) have all coefficients of the higher degrees ending with the $ \left[\frac{k}{2}\right] $ degree in common. Moreover, we will prove that, for a fixed value of $k$, all the first, second, etc. coefficients of the common part of the fractal family have a general form, showing the vertical connection between the corresponding coefficients of all fractal family $\left\{ p\left(n, k\right), k\in \mathbb{N}\right\} $. Furthermore, for a fixed value of $k$, all the coefficients within the same polynomial have a unique general form, showing the horizontal connection of the coefficients of the polynomial $p\left(n, k\right)$. The partition function is not real a polynomial, but it can be written as a fractal polynomial which can be obtained from the general form of the partition class functions $\left\{p\left(n, k\right)\right\}$. In that case, the partition function for each $n$ uses a different polynomial. We show that all these polynomials can be combined with one single in which each member can be a formula for calculating the total number of partitions of all natural numbers.

Sequentially Congruent Partitions and Related Bijections

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict "frequency congruence" condition -- the frequency (or multiplicity) of each part is divisible by that part -- and prove families of similar bijections, connecting with G. E. Andrews's theory of partition ideals.

Efficient discretisation of stochastic differential equations

The aim of this study is to find a generic method for generating a path of the solution of a given stochastic differential equation which is more efficient than the standard Euler–Maruyama scheme with Gaussian increments. First we characterize the asymptotic distribution of pathwise error in the Euler–Maruyama scheme with a general partition of time interval and then, show that the error is reduced by a factor when using a partition associated with the hitting times of sphere for the driving d-dimensional Brownian motion. This reduction ratio is the best possible in a symmetric class of partitions. Next we show that a reduction which is close to the best possible is achieved by using the hitting time of a moving sphere that is easier to implement.

Partitioning the Power Set of $[n]$ into $C_k$-free parts

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.

Calculating an upper bound of the locating-chromatic number of trees

The locating-chromatic number of a graph G(V,E) is the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices not contained in the same partition class. Determining the locating-chromatic number of any tree is a difficult task. In this paper, we propose an algorithm to compute the upper bound on the locating-chromatic number of any tree. To do so, we decompose a tree into caterpillars and then compute the upper bound of the locating-chromatic number of this tree in terms of the ones for these caterpillars.

Locating Chromatic Number of Powers of Paths and Cycles

Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (2009) [14] for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangular or double-staircase shapes and all of the variables are set to 1, then factorization identities for numbers of certain plane partitions, alternating sign matrices and related combinatorial objects are obtained.

Minimum codegree condition for perfect matchings in k‐partite k‐graphs

AbstractLet be a ‐partite ‐graph with vertices in each partition class, and let denote the minimum codegree of . We characterize those with and with no perfect matching. As a consequence, we give an affirmative answer to the following question of Rödl and Ruciński: if is even or , does imply that has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of ‐sets.

Partitioning two‐coloured complete multipartite graphs into monochromatic paths and cycles

AbstractWe show that any complete ‐partite graph on vertices, with , whose edges are two‐coloured, can be covered with two vertex‐disjoint monochromatic paths of distinct colours, given that the largest partition class of contains at most vertices. This extends known results for complete and complete bipartite graphs.Secondly, we show that in the same situation, all but vertices of the graph can be covered with two vertex‐disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded. From this we derive that the whole graph, if large enough, may be covered with 14 vertex‐disjoint monochromatic cycles.

On the Locating Chromatic Number of Certain Barbell Graphs

The locating chromatic number of a graph G is defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. In this paper we investigate the locating chromatic number for two families of barbell graphs.

Limit Shapes via Bijections

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

Combinatorial interpretations of Ramanujan’s tau function

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.

Integer Partitions with Even Parts Below Odd Parts and the Mock Theta Functions

The paper begins with a study of a couple of classes of partitions in which each even part is smaller than each odd. In one class, a Dyson-type crank exists to explain a mod 5 congruence. The second part of the paper treats the arithmetic and combinatorial properties of the third order mock theta function $${\nu(q)}$$ and relates the even part of $${\nu(q)}$$ to the partitions initially considered. We also consider a surprisingly simple combinatorial relationship between the cranks and the ranks of the partition of n.

Counting Binomial Coefficients Divisible by a Prime Power

We define a new class of integer partitions generalizing hyperbinary partitions. We then utilize these partitions to provide a new, relatively simple formula for the number of binomial coefficients in a particular row of Pascal’s triangle that are divisible by a fixed power of a prime. Our formula specializes to a famous result due to N. J. Fine from 1947. Furthermore, we demonstrate a connection between the binomial coefficients congruent to zero modulo a fixed prime power and the so-called digital dominance orders on the natural numbers.

On the honeycomb conjecture for a class of minimal convex partitions

We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every n ∈ N n \in \mathbb {N} the minimizers of F F among convex n n -gons with given area. In particular, our result allows us to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover, we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of λ 1 \lambda _1 among convex pentagons, hexagons, and heptagons with prescribed area.

A transmission problem on a polygonal partition: regularity and shape differentiability

ABSTRACTWe consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes of the partition. This allows to prove shape differentiability of solutions and to establish an explicit formula for the shape derivative.