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.     

Fourth-Order p-Laplacian Nonlinear Systems via the Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem

In this paper, existence results for a fourth-order nonlinear system are obtained. Both classical and vector versions of the Krasnosel’skiĭ’s fixed point theorem are used and a comparison of the obtained results to those from the literature is provided.     

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.     

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 $$a=0$$...     

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.     

Partially Ordered Generalized Patterns and <Emphasis Type="Italic">k</Emphasis>-ary Words

Recently, Kitaev [9] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrímsson [1]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p with no dashes (which is allowed to have repetition of letters), provided we know the g.f. for the number of k-ary words that avoid p.AMS Subject Classification: 05A05, 05A15.     

Restricted even permutations and Chebyshev polynomials

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.