Advances in Applied Mathematics最新文献

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.

We present a computational algebra solution to reverse engineering the network structure of discrete dynamical systems from data. We use pseudomonomial ideals to determine dependencies between variables that encode constraints on the possible wiring diagrams underlying the process generating the discrete-time, continuous-space data. Our work assumes that each variable is either monotone increasing or decreasing. We prove that with enough data, even in the presence of small noise, our method can reconstruct the correct unique wiring diagram.

The number of inversion sequences avoiding two patterns 101 and 102 is known to be the same as the number of permutations avoiding three patterns 2341, 2431, and 3241. This sequence also counts the number of Schröder paths without triple descents, restricted bicolored Dyck paths, $(101,021)$-avoiding inversion sequences, and weighted ordered trees. We provide bijections to integrate them together by introducing F-paths. Moreover, we define three kinds of statistics for each of the objects and count the number of each object with respect to these statistics. We also discuss direct sums of each object.

We introduce two new bases of $QSym$, the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in NSym, the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions ρ and ω, which generalize the classical involution ω on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and $QSym$ respectively. We then use the involutions ρ and ω to translate these and other known results to our reverse and row-strict reverse bases.