In this paper, we investigate properties of a new class of generalized Cauchy numbers. By using the method of coecient, we establish a series of identities involving generalized Cauchy numbers, which generalize some results for the Cauchy numbers. Furthermore, we give some asymptotic approximations of certain sums related to the generalized Cauchy numbers.

Let \( F(x) = \sum_{\nu \in \mathbb{N}^d} F_\nu x^\nu \) be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume \( F = G / H \) for some functions \( G \) and \( H \) holomorphic in a neighborhood of the origin. We derive asymptotics for the coefficients \( F_{r\alpha} \) as \( r \to \infty \) with \( r\alpha \in \mathbb{N}^d \) for \( \alpha \) in a permissible subset of \( d \)-tuples of positive reals. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for \( F_{r\alpha} \) when the asymptotics are controlled by a transverse multiple point of the analytic variety \( H = 0 \). This improves upon earlier work by R. Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and apply it to obtain accurate numerical results for several rational combinatorial generating functions.

Let \( P(n, k) \) denote the set of partitions of \( [n] = \{1, 2, \ldots, n\} \) containing exactly \( k \) blocks. Given a partition \( \Pi = B_1 / B_2 / \cdots / B_k \in P(n, k) \) in which the blocks are listed in increasing order of their least elements, let \( \pi = \pi_1 \pi_2 \cdots \pi_n \) denote the canonical sequential form wherein \( j \in B_{\pi_j} \) for all \( j \in [n] \). In this paper, we supply an explicit formula for the generating function which counts the elements of \( P(n, k) \) according to the number of strings \( k1 \) and \( r(r+1) \), taken jointly, occurring in the corresponding canonical sequential forms. A comparable formula for the statistics on \( P(n, k) \) recording the number of strings \( 1k \) and \( r(r-1) \) is also given, which may be extended to strings \( r(r-1) \cdots (r-m) \) of arbitrary length using linear algebra. In addition, we supply algebraic and combinatorial proofs of explicit formulas for the total number of occurrences of \( k1 \) and \( r(r+1) \) within all the members of \( P(n, k) \).

A word is centrosymmetric if it is invariant under the reverse-complement map. In this paper, we give  enumerative results on k-ary centrosymmetric words of length n avoiding a pattern of length 3 with no repeated letters.

We consider a bivariate rational generating function
\[
F(x, y) = \frac{P(x, y)}{Q(x, y)} = \sum_{r, s \geq 0} a_{r,s} x^r y^s
\]
under the assumption that the complex algebraic curve \( \mathcal{V} \) on which \( Q \) vanishes is smooth. Formulae for the asymptotics of the coefficients \( \{a_{r,s}\} \) are derived in [PW02]. These formulae are in terms of algebraic and topological invariants of \( \mathcal{V} \), but up to now these invariants could be computed only under a minimality hypothesis, namely that the dominant saddle must lie on the boundary of the domain of convergence. In the present paper, we give an effective method for computing the topological invariants, and hence the asymptotics of {\(a_{rs}\)}, without the minimality assumption. This leads to a theoretically rigorous algorithm, whose implementation is in progress at http://www.mathemagix.org

This paper presents a new construction of the \( m \)-fold metaplectic cover of \( \mathrm{GL}_n \) over an algebraic number field \( k \), where \( k \) contains a primitive \( m \)-th root of unity. A 2-cocycle on \( \mathrm{GL}_n(\mathbb{A}) \) representing this extension is given, and the splitting of the cocycle on \( \mathrm{GL}_n(k) \) is found explicitly. The cocycle is smooth at almost all places of \( k \). As a consequence, a formula for the Kubota symbol on \( \mathrm{SL}_n \) is obtained. The construction of the paper requires neither class field theory nor algebraic \( K \)-theory but relies instead on naive techniques from the geometry of numbers introduced by W. Habicht and T. Kubota. The power reciprocity law for a number field is obtained as a corollary.

Let \( \pi = \pi_1 \pi_2 \cdots \pi_n \) be any permutation of length \( n \), we say a descent \( \pi_i \pi_{i+1} \) is a {lower}, {middle}, {upper} if there exists \( j > i+1 \) such that \( \pi_j < \pi_{i+1}, \pi_{i+1} < \pi_j < \pi_i, \pi_i < \pi_j \), respectively. Similarly, we say a rise \( \pi_i \pi_{i+1} \) is a {lower}, {middle}, {upper} if there exists \( j > i+1 \) such that \( \pi_j < \pi_i, \pi_i < \pi_j < \pi_{i+1}, \pi_{i+1} < \pi_j \), respectively. In this paper, we give an explicit formula for the generating function for the number of permutations of length \( n \) according to the number of upper, middle, lower rises, and upper, middle, lower descents. This allows us to recover several known results in the combinatorics of permutation patterns as well as many new results. For example, we give an explicit formula for the generating function for the number of permutations of length \( n \) having exactly \( m \) middle descents.

We prove that a sumset of a TE subset of N (these sets can be viewed as “aperiodic” sets) with a set of positive upper density intersects any polynomial sequence. For WM sets (subclass of TE sets) we prove that the intersection has lower Banach density one. In addition we obtain a generalization of the latter result to the case of several polynomials.

In this paper, we prove the Tiling implies Spectral part of Fuglede’s cojecture for the three interval case. Then we prove the converse Spectral implies Tiling in the case of three equal intervals and also in the case where the intervals have lengths 1/2, 1/4, 1/4. Next, we consider a set Ω ⊂ R, which is a union of n intervals. If Ω is a spectral set, we prove a structure theorem for the spectrum provided the spectrum is assumed to be contained in some lattice. The method of this proof has some implications on the Spectral implies Tiling part of Fuglede’s conjecture for three intervals. In the final step in the proof, we need a symbolic computation using Mathematica. Finally with one additional assumption we can conclude that the Spectral implies Tiling holds in this case.

We show that if \( A \) is a finite subset of an abelian group with additive energy at least \( c|A|^3 \), then there is a set \( \mathcal{L} \subset A \) with \( |\mathcal{L}| = O(c^{-1} \log |A|) \) such that \( |A \cap \mathrm{Span}(\mathcal{L})| = \Omega(c^{1/3} |A|) \).