Formation of Structured Polygonal Nanoparticles by Phase‐Separated Comb‐Like Polymers

A simple approach using comb‐like polymers that undergo nanophase separation between the polyester backbone and the stearoyl side chains is proposed for the preparation of structured non‐spherical nanoparticles from a nanoemulsion. Depending on the degree of esterification of the OH groups of poly(glycerol adipate) differently ordered nanostructures is obtained. A perfect lamellar arrangement is obtained for polymers with a high degree of esterification and leads to spherical nanoparticles with an internal onion‐like structure. However, when the degree of esterification is only 20 mol%, polygonal nanoparticles with an internal pseudo‐hexagonal structure are obtained. The differences in the nanoparticle shapes are related to the volume fraction of the paraffinic pool.     

A combinatorial proof of a result for permutation pairs

In this paper, a direct combinatorial proof is given of a result on permutation pairs originally due to Carlitz, Scoville, and Vaughan and later extended. It concerns showing that the series expansion of the reciprocal of a certain multiply exponential generating function has positive integer coefficients. The arguments may then be applied to related problems, one of which concerns the reciprocal of the exponential series for Fibonacci numbers.     

Separable d-Permutations and Guillotine Partitions

We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula, and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between separable d-dimensional permutations and guillotine partitions of a 2 ^d-1-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain several results on patterns in permutations.     

A polynomial generalization of some associated sequences related to set partitions

Periodica Mathematica Hungarica - In this paper, we consider a common polynomial generalization, denoted by $$w_m(n,k)=w_m^{a,b,c,d}(n,k)$$ , of several types of associated sequences. When $$a=0$$...     

Several variants of the Dumont differential system and permutation statistics

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont(1979)and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications,we present some characterizations for several permutation statistics.     

Ascents of size less than <Emphasis Type="Italic">d</Emphasis> in compositions

A composition of a positive integer n is a finite sequence π₁π₂...π _m of positive integers such that π₁+...+π _m = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if π _i+1 ≥ π _i +d (respectively, π _i < π _i+1 < π _i + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable generating function for the number of compositions of n according to the number of parts, ascents of size d or more, ascents of size less than d, descents and levels. Also, we extend the results of Brennan and Knopfmacher to the case of ascents of size less than d. More precisely, we determine the mean, variance and limiting distribution of the number of ascents of size less than d in the set of compositions of n.     

Restricted partitions and q-Pell numbers

In this paper, we provide new combinatorial interpretations for the Pell numbers p _n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p _n . By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p _n . Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.     

Pattern avoiding partitions and Motzkin left factors

Let L _n , n ≥ 1, denote the sequence which counts the number of paths from the origin to the line x = n ? 1 using (1, 1), (1, ?1), and (1, 0) steps that never dip below the x-axis (called Motzkin left factors). The numbers L _n count, among other things, certain restricted subsets of permutations and Catalan paths. In this paper, we provide new combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify four classes of the partitions of size n, all of which have cardinality L _n and each avoiding a set of two classical patterns of length four. We obtain a further generalization in one of the cases by considering a pair of statistics on the partition class. In a couple of cases, to show the result, we make use of the kernel method to solve a functional equation arising after a certain parameter has been introduced.     

Restricted 1-3-2 Permutations and Generalized Patterns

Recently, Babson and Steingrimsson (see [2]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 1-3-2 (or containing 1-3-2 exactly once) and an arbitrary generalized pattern t \tau on k letters, or containing t \tau exactly once. In several cases, the generating function depends only on k and can be expressed via Chebyshev polynomials of the second kind, and the generating function of Motzkin numbers.     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.