Context-free grammars,generating functions and combinatorial arrays

Let ${\left[R_{n,k}\right]}_{n,k\ge 0}$ be an array of nonnegative numbers satisfying the recurrence relation $R_{n,k}=(a_{1}n+a_{2}k+a_3)R_{n-1,k}+(b_{1}n+b_{2}k+b_3)R_{n-1,k-1}+(c_{1}n+c_{2}k+c_3)R_{n-1,k-2}$ with $R_{0,0}=1$ and $R_{n,k}=0$ unless $0\le k\le n$. In this paper, we first prove that the array ${\left[R_{n,k}\right]}_{n,k\ge 0}$ can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of ${\left[R_{n,k}\right]}_{n,k\ge 0}$. Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type $B$, and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong $q$-log-convexity of some generating functions.     

Digit sums and generating functions

The Ramanujan Journal - We connect a primitive operation from arithmetic—summing the digits of a base-B integer—to q-series and product generating functions analogous to those in...     

Haruspicy and anisotropic generating functions

Guttmann and Enting [Phys. Rev. Lett. 76 (1996) 344–347] proposed the examination of anisotropic generating functions as a test of the solvability of models of bond animals. In this article we describe a technique for examining some properties of anisotropic generating functions. For a wide range of solved and unsolved families of bond animals, we show that the coefficients of yⁿ is rational, the degree of its numerator is at most that of its denominator, and the denominator is a product of cyclotomic polynomials. Further, we are able to find a multiplicative upper bound for these denominators which, by comparison with numerical studies [Jensen, personal communication; Jensen and Guttmann, personal communication], appears to be very tight. These facts can be used to greatly reduce the amount of computation required in generating series expansions. They also have strong and negative implications for the solvability of these problems.     

Polynomial expansions and generating functions

We study expansions in polynomials {P_n(x)}^∞_o generated by ∑^∞_{n = o}P_n(x)tⁿ = A(t) φ(xt^kθ(t)), θ(0) ≠ 0, and ∑^∞_{n = 0}P_n(x)tⁿ = ∑^k_{j = 1}A_j(t) φ(xt?_j), ?₁,…,?_k being the k roots of unity. The case k = 1 is contained in a recent work by Fields and Ismail. We also prove a new generalization of Vandermond's inverse relations.     

Uniform partition extensions,a generating functions perspective

In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths are defined. It is proved that the function CF(x, y) is a generating function defined on some rooted lattice paths with end point on y = 1. So, by a simple and unified method, from the view of lattice paths, we obtain two combinatorial interpretations of this bivariate function and derive two uniform partitions on these rooted lattice paths.     

ContributionsFinitary bases and formal generating functions

Let k be a nonzero, commutative ring with 1, and let R be a k-algebra with a countably-infinite ordered free k-basis B = [p_n: n 0]. We characterize and analyze those bases from which one can construct a k-algebra of ‘formal B-series’ of the form f=∑c_n p_n, with c_n ε k, showing inter alia that many classical polynomial bases fail to have this property.     

Simple permutations and algebraic generating functions

A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating subsets that are restricted by properties belonging to a finite “query-complete set.” Such properties include being even, being an alternating permutation, and avoiding a given generalised (blocked or barred) pattern. We show that the generating functions for these subsets are always algebraic, thereby generalising recent results of Albert and Atkinson. We also apply these techniques to the enumeration of involutions and cyclic closures.     

Bilateral generating functions and operational methods

Bilateral generating functions are those involving products of different types of polynomials. We show that operational methods offer a powerful tool to derive these families of generating functions. We study cases relevant to products of Hermite polynomials with Laguerre, Legendre and other polynomials. We also propose further extensions of the method which we develop here.     

Finitary bases and formal generating functions

Let k be a nonzero, commutative ring with 1, and let R be a k-algebra with a countably-infinite ordered free k-basis B = [p_n: n 0]. We characterize and analyze those bases from which one can construct a k-algebra of ‘formal B-series’ of the form f=∑c_np_n, with c_n ε k, showing inter alia that many classical polynomial bases fail to have this property.     

Belief functions on real numbers

We generalize the TBM (transferable belief model) to the case where the frame of discernment is the extended set of real numbers , under the assumptions that ‘masses’ can only be given to intervals. Masses become densities, belief functions, plausibility functions and commonality functions become integrals of these densities and pignistic probabilities become pignistic densities. The mathematics of belief functions become essentially the mathematics of probability density functions on .     

Symplectic homology via generating functions

Research at Mathematical Sciences Research Institute supported by an MSRI postdoctoral fellowship. Research at Stanford supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. Research at Centre Emile Borel, Institut Henri Poincaré, Paris funded by Ministère des Affaires Etrangères, France.     

Belief functions contextual discounting and canonical decompositions

In this article, the contextual discounting of a belief function, a classical discounting generalization, is extended and its particular link with the canonical disjunctive decomposition is highlighted. A general family of correction mechanisms allowing one to weaken the information provided by a source is then introduced, as well as the dual of this family allowing one to strengthen a belief function.     

Lattice point generating functions and symmetric cones

We show that a recent identity of Beck–Gessel–Lee–Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.     

Asymptotics and algebraicity of some generating functions

Some generating functions ∑_{n ⩾ 0}f(n) xⁿ, arising in combinatorics and algebra, are shown to be nonalgebraic by calculating the asymptotics of f(n). Some other such generating functions are shown to be algebraic by an application of determinantal varieties and G-invariant ideals.     

Clusters,generating functions and asymptotics for consecutive patterns in permutations

We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of lengths 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function.We also complete the classification of consecutive patterns of length up to 6 into equivalence classes, proving a conjecture of Nakamura. Finally, we show that the monotone pattern asymptotically dominates (in the sense that it is easiest to avoid) all non-overlapping patterns of the same length, thus proving a conjecture of Elizalde and Noy for a positive fraction of all patterns.     

On generating functions of multiple zeta values and generalized hypergeometric functions

Generating functions for sums of certain multiple zeta values with fixed weight, depth and i-heights are discussed. The functions are systematically expressed in terms of generalized hypergeometric functions. The expressions reproduce several known formulas for multiple zeta values as applications.     

Ramanujan-type congruences for certain generating functions

For nonnegative integers a, b, the function d _a,b (n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$ . We establish some Ramanujan-type congruences for d _a,b (n) by the theory of modular forms with complex multiplication. As consequences, we generalize the famous Ramanujan congruences for the partition function p(n) modulo 5, 7, and 11.