On the distribution of the summands of unequal partitions in residue classes

Summary It is proved that the summands of almost all unequal partitions of nare well-distributed modulo dfor d=o(n^1/2).     

Divisors in residue classes, constructively

Let be integers satisfying , , , and let . Lenstra showed that the number of integer divisors of equivalent to is upper bounded by . We re-examine this problem, showing how to explicitly construct all such divisors, and incidentally improve this bound to .

     

The distribution of sequences in residue classes

We prove that any set of integers with lies in at least many residue classes modulo most primes . (Here is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below which are multiplicatively generated by the coprime integers (i.e. whose counting function is also ) lie in at least residue classes, modulo most small primes , where as .

     

Counting primes in residue classes

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing can be used for computing efficiently , the number of primes congruent to modulo up to . As an application, we computed the number of prime numbers of the form less than for several values of up to and found a new region where is less than near .

     

Arithmetic Properties of Summands of Partitions

Let d d, d2 2. We prove that for almost all partitions of an integer the parts are well distributed in residue classes mod d. The limitations of the uniformity of this distribution are also studied.     

Arithmetic Properties of Summands of Partitions II

Let d∈ℕ, d ≥ 2. We prove that a positive proportion of partitions of an integer n satisfies the following : for all 1≤ a < b≤ d, the number of the parts congruent to a (mod d) is greater than the number of the parts congruent to b (mod d). We also show that for almost all partitions the rate of the number of square free parts is . 2000 Mathematics Subject Classification: Primary—11P82     

Gibbs partitions: The convergent case

We study Gibbs partitions that typically form a unique giant component. The remainder is shown to converge in total variation toward a Boltzmann‐distributed limit structure. We demonstrate how this setting encompasses arbitrary weighted assemblies of tree‐like combinatorial structures. As an application, we establish smooth growth along lattices for small block‐stable classes of graphs. Random graphs with n vertices from such classes are shown to form a giant connected component. The small fragments may converge toward different Poisson Boltzmann limit graphs, depending along which lattice we let n tend to infinity. Since proper addable minor‐closed classes of graphs belong to the more general family of small block‐stable classes, this recovers and generalizes results by McDiarmid (2009).     

Pairing conjugate partitions by residue classes

We study integer partitions in which the parts fulfill the same congruence relations with the parts of their conjugates, called conjugate-congruent partitions. The results obtained include uniqueness criteria, weight lower-bounds and enumerating generating functions.     

On zero-sum partitions and anti-magic trees

We study zero-sum partitions of subsets in abelian groups, and apply the results to the study of anti-magic trees. Extension to the nonabelian case is also given.     

Double perfect partitions

MacMahon [Combinatory Analysis, vols. I and II, Cambridge University Press, Cambridge, 1915, 1916 (reprinted, Chelsea, 1960)] introduced a perfect partition of positive integer n, which is a partition such that every positive integer less than or equal to n can be uniquely represented by the sum of its parts. We generalize perfect partition and find a relation with ordered factorizations.     

On Kestenband–Ebert partitions

We study Kestenband–Ebert partitions from a group-theoretic point of view. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 367–375, 1997     

On the number of partitions with distinct even parts

Let ped(n) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for ped(n)mod2 and mod4 by the theory of Hecke eigenforms.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

Multiplicity of summands in the random partitions of an integer

In this paper, we prove a conjecture of Yakubovich regarding limit shapes of ‘slices’ of two-dimensional (2D) integer partitions and compositions of n when the number of summands m ~An ^α for some A?>?0 and $\alpha < \frac{1}{2}$ . We prove that the probability that there is a summand of multiplicity j in any randomly chosen partition or composition of an integer n goes to zero asymptotically with n provided j is larger than a critical value. As a corollary, we strengthen a result due to Erdös and Lehner (Duke Math. J. 8 (1941) 335–345) that concerns the relation between the number of integer partitions and compositions when $\alpha = \frac{1}{3}$ .     

On certain unimodal sequences and strict partitions

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.     

Polychromatic colorings of rectangular partitions

A rectangular partition is a partition of a plane rectangle into an arbitrary number of non-overlapping rectangles such that no four rectangles share a corner. In this note, it is proven that every rectangular partition admits a vertex coloring with four colors such that every rectangle, except possibly the outer rectangle, has all four colors on its boundary. This settles a conjecture of Dinitz et al. [Y. Dinitz, M.J. Katz, R. Krakovski, Guarding rectangular partitions, in: Abstracts 23rd Euro. Workshop Comput. Geom., 2007, pp. 30-33]. The proof is short, simple and based on 4-edge-colorability of a specific class of planar graphs.     

Nodal domains and spectral minimal partitions

We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between the nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions and the characterization of the minimal partitions associated with nodal sets as the nodal domains of Courant-sharp eigenfunctions.