Advances in Applied Mathematics最新文献

We study, through new recurrence relations for certain binomial coefficients modulo a power of a prime, the evolution of the iterated anti-differences of periodic sequences modulo m. We prove that one can reduce to study iterated anti-differences of constant sequences. Finally we apply our results to describe the dynamics of the iterated applications of the Vieru operator to the sequence considered by the Romanian composer Vieru in his Book of Modes [20].

MacMahon's classical theorem on the number of boxed plane partitions has been generalized in several directions. One way to generalize the theorem is to view boxed plane partitions as lozenge tilings of a hexagonal region and then generalize it by making some holes in the region and counting its tilings. In this paper, we provide new regions whose numbers of lozenges tilings are given by simple product formulas. The regions we consider can be obtained from hexagons by removing structures called intrusions. In fact, we show that the tiling generating functions of those regions under certain weights are given by similar formulas. These give the q-analogue of the enumeration results.

Binary search trees (BST) are a popular type of structure when dealing with ordered data. They allow efficient access and modification of data, with their height corresponding to the worst retrieval time. From a probabilistic point of view, BSTs associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform permutation.

We consider here the case where the input comes from i.i.d. random points in the plane with law μ, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree only depends on the behavior of the measure μ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures μ. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.

A pairing on an arbitrary ground set Ω is a triple $P:=(U,F,\Lambda )$, with $U,\Lambda$ two sets and $F:U\times \Omega ⟶\Lambda$ a map. Several properties of pairings arise after considering the Moore set system $M_P$ and the abstract simplicial complex $N_P$ on Ω, defined by taking the maximum and the minimal elements of the equivalence collections with respect to a specific equivalence relation ${\approx }_P$, respectively called minimal and maximum partitioners.

In the present work we first detect various sufficient conditions allowing us to represent specific subfamilies of abstract simplicial complexes as the family of all the minimal partitioners of some pairing on the same ground set. Next, we classify two suitable subcollections of pairings by using generalized matroidal-like properties of $N_P$. More in detail, we first determine a sufficient condition on $P$ ensuring that the family $N_P$ is a closable finitary simplicial complex and call the resulting pairings attractive. On an arbitrary ground set Ω, attractiveness, together with a finiteness condition, implies that the minimal members of the equivalence collections of each $X\in M_P$ with respect to ${\approx }_P$ all have the same cardinality. Nevertheless, the converse does not hold, neither in the finite case. To this regard, we find some counterexamples inducing us to introduce the class of quasi-attractive pairings. We carried out a detailed analysis of quasi-attractive pairings: for instance we characterize them from a lattice-theoretic point of view and, on a finite ground set Ω, also in term of exchange properties of suitable set systems.

Finally, by taking the adjacence matrix of a simple undirected graph G as a model of pairing, we show that the Petersen graph induces an attractive pairing, while the Erdös' friendship graphs induce a quasi-attractive, but not attractive, one.

The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. In order to extend a result of Diaconis et al. (2014) [16], we show that two triple set-valued statistics of permutations are equidistributed. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc $(p,q)$-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomial.

In this paper, we consider two matrices of polynomials ${\left[h_{n,k}\right(t\left)\right]}_{n\ge 0}$ and ${\left[l_{n,k}\right(t\left)\right]}_{n\ge 0}$, which are defined by a recurrence relation from a sequence of polynomials with real coefficients. These matrices play an important role in the study of the Eulerian and derangement transformations by Athanasiadis, who asked when their rows form interlacing sequences of real-rooted polynomials. In this paper, we give an answer to this question. In our setup, all columns of these matrices are shown to be generalized Sturm sequences. As applications, we show that the derangement transformation, its type B analogue and the r-colored derangement transformation of a class of polynomials with nonnegative coefficients, and all roots in the interval $[-1,0]$, have only real roots in a unified manner. The question about the type B derangement transformation was also raised by Athanasiadis. Furthermore, we show that the diagonal line of ${\left[h_{n,k}\right(t\left)\right]}_{n\ge 0}$ and ${\left[l_{n,k}\right(t\left)\right]}_{n\ge 0}$ forms a generalized Sturm sequence respectively, i.e., we give sufficient conditions for the binomial transformation to preserve the interlacing property.

Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper we show that the support of each dual k-Schur polynomial indexed by a k-bounded partition coincides with that of the Schur polynomial indexed by the same partition, and hence the two polynomials share the same saturated Newton polytope. The main result is based on our recursive algorithm to generate a semistandard k-tableau for a given shape and k-weight. As consequences, we obtain the M-convexity of dual k-Schur polynomials, affine Stanley symmetric polynomials and cylindric skew Schur polynomials.

A D-permutation is a permutation of $\left[2n\right]$ satisfying $2k-1\le \sigma (2k-1)$ and $2k\ge \sigma \left(2k\right)$ for all k; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.

Tandem duplication random loss (TDRL) and inverse tandem duplication random loss (iTDRL) are mechanisms of mitochondrial genome rearrangement that can be modeled as simple operations on signed permutations. Informally, they comprise the duplication of a subsequence of a permutation, where in the case of iTDRL the copy is inserted with inverted order and signs. In the second step, one copy of each duplicate element is removed, such that the result is again a signed permutation. The TDRL/iTDRL sorting problem consists in finding the minimal number of TDRL or iTDRL operations necessary to convert the identity permutation ι into a given permutation π. We introduce a simple signature, called the misc-encoding, of permutation π. This construction is used to design an $O(n\log n)$ algorithm to solve the TDRL/iTDRL sorting problem.