Marilena Barnabei

Recursive matrices and umbral calculus

Two major objectives can be seen to guide much recent work in enumeration: (1) to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide-enough type; (2) to recover from empirical data an underlying set-theoretic structure which would reveal the source of the given recursion. We are here concerned with the first of these objectives, though the eventual understanding of the second is tacitly present, if only as a goal. We noticed the coincidence of several computations which, similar as they are in retrospect, had failed to realize their kinship. On leafing through the unique assembly of recursively solvable combinatorial problems in Comtet’s and Sloane’s invaluable collections, one is struck by the repeated occurrence of one and the same kind of double recursion. More strikingly, the same recursion is seen to occur in the polynomial sequences of the Umbra1 Calculus of Roman and Rota (see [25]). Everywhere, the Lagrange inversion formula for power series plays a pivotal role. Much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula, both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs (Andrews, Foata, Garsia, Gessel, Joni, Raney, Reiner, Schtitzenberger, to name but a few). Whatever their origins, the identities abutting Lagrange inversion are expressed by integers alone. This suggests not only a hidden set-theoretic layer, but a characteristic-free generalization as well: this generalization is the central theme of our work. We define a monoid of infinite matrices-“recursive matrices” for short. The entries of these matrices give the sought-out recursion, for example, that for coefficients of binomial and Sheffer polynomials and factor sequences, as well as that of the special sequences recently introduced by Roman in [26]. 546 002 l-8693/82/040546-28%02.00/‘0

Polynomial sequences of integral type

Such sequences of polynomials have been intensively studied in recent years (see, e.g., [2, 4-6, S-101). Our point of view in this work will be to assume these sequences to be integer-valued. This assumption, far from being a restriction, leads to several unexpected simplifications in the theory and brings it closer to its combinatorial sources. We have therefore decided to develop the theory anew along these lines, that is, by working exclusively over the ring of integers. While some of the results relating to the operator calculus associated therewith may be gleaned from analogous results in the umbra1 calculus, there is nontheless a body of results which make sequences of integral type (as we shall call them) strikingly different in their behavior. To wit, a sequence of polynomials of integral type is uniquely associated with a shift-invariant operator Q-the delta operator of the umbra1 calculus-which in the present case can be uniquely expanded into a formal power series of the difference operator A:

Recursive properties of Toeplitz and Hurwitz matrices

Abstract Banded Toeplitz and Hurwitz matrices are shown to be particular cases of a more general class of biinfinite matrices, called recursive matrices. The main features of Toeplitz and Hurwitz matrices can thereby be seen to be immediate consequences of a fundamental theorem about recursive matrices, called the product rule. Moreover, some properties of products of Toeplitz and Hurwitz matrices can be proved by similar arguments. Some applications related to the general theory of compactly supported wavelets are presented.

Some algebraic aspects of signal processing

Abstract It has recently been shown in (M. Barnabei, L.B. Montefusco, Linear Algebra and applications 274 (1998) 367–388) that the algebraic-combinatorial notion of recursive matrix can fruitfully be used to represent and easily handle the basic operations of filter theory, such as convolution, up-sampling, and down-sampling. In this paper we show how the recursive matrix reinterpretation of two-channel FIR filter bank theory leads to a notable simplification in language and proofs, together with an easy and immediate generalization to the M-channel case. For example, in both 2-channel and M-channel cases, perfect reconstruction and alias concelation conditions can be restated in an algebraic language, thereby obtaining an easy and constructive proof using the fundamental properties of recursive matrices.

Descent sets on 321-avoiding involutions and hook decompositions of partitions

We show that the distribution of the major index over the set of involutions in S n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside a ? n 2 ? × ? n 2 ? box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.

On small {;k; q};-arcs in planes of order q2

In this paper we examine some properties of complete {;k; q};-arcs in projective planes of order q2. In particular, we derive a lower bound for k, and we exhibit a family of arcs having low values of k which exist in every such plane having a Baer subplane. In addition we resolve the existence problem for complete {;k; 3};-arcs in PG(2, 9).

Circulant recursive matrices

Recursive matrices are bi-infinite matrices which can be recursively generated starting from two given Laurent series α and β, called the recurrence rule and the boundary value of the matrix, respectively. The i th row of a recursive matrix contains the coefficients of the series α i β. These matrices were introduced by Barnabei, Brini, and Nicoletti [1] to formulate a more general version of the umbral calculus developed in a series of fundamental papers by Rota and his school [7, 8, 9]. In [2] it was shown that the most important operators of the umbral calculus can be represented by recursive matrices. In particular, shift-invariant operators, which play a crucial role in Rota’s theory, correspond to Toeplitz matrices, which can be characterized as recursive matrices with recurrence rule α(t) = t. Recently, recursive matrices were found to be useful in studying algebraic aspects of signal processing, since they contain the classes of bi-infinite Toeplitz, Hankel and Hurwitz matrices as special cases [3, 4].

The Littlewood-Richardson rule for Coschur modules

In the last 15 years, many mathematicians have turned their attention to the problem of extending to more general settings (fields of arbitrary characteristic, commutative rings) the representation theory of the general linear and symmetric groups over fields of characteristic zero. In this undertaking, the first problem is, perhaps, that of finding suitable generalizations of those modules that, in the classical theory, yield the irreducible representations first discovered and classified by Schur in his Dissertation of 1901 [24]. Over fields of characteristic zero, the more popular constructions of these modules, in terms of symmetry classes of tensors, are those due to Schur [25] and Weyl [31]; as is well known, these contructions are equivalent since they give rise to modules that, even though they are different as subspaces of the tensors space, turn out to be isomorphic with respect to the action of the group. Recently, it has been recognized that such constructions can be adequately adapted in order to make sense over arbitrary commutative rings; Akin, Buchsbaum and Weyman [ 1 ] have, therefore, introduced the so-called Schur and Coschur functors that turn out to be the natural generalizations to commutative rings of the constructions of Schur and Weyl, respectively. These functors are universally free functors and they give rise to modules-Schur and Coschur modules-which, even though they are isomorphic over fields of characteristic zero, are far from being isomorphic over the integers. Furthermore, Schur and Coschur modules are indecomposable, but they are not, in general, irreducible over commutative rings. Working from a slightly different point of view, Carter and Lusztig [7] discovered a class of modules which are irreducible over fields of positive characteristic; these modules were later elegantly described by Clausen [S] as the images of a natural operator-the Capelli operator-whose domain and codomain are the so-called Weyl modules of the second and first kind, which turn out to be the same as Coschur and Schur modules, respectively. Alongside the classification of the irreducible modules lies the second 143

On arcs with weighted points

Abstract In the present paper we start the study of arcs with weighted points in a finite projective plane, and we get some particular results about the existence or non-existence of these arcs.

Lagrange inversion in infinitely many variables

This paper is concerned with the problem of finding an effective definition for the compositional inverse of formal power series in several variables, with special regard to series in infinitely many variables. The problem of defining analogs for several variables of the Lagrange inversion formula has been open for a long time (see, e.g., [3, 53). So far, all indications point to the fact that an explicit Lagrange inversion formula in infinitely many variables does not exist in the complete generality that we find for series in finitely many variables [4]. Our main result in this paper is a Lagrange inversion formula for sets of power series in infinitely many variables c~r(xr, x2 ,... ), c(~(x~, x2 ,...) ,..., where the formal power series X,(X,, x,,...) does not depend on the variables Xl, x27.7 x, , * This situation occurs in several instances, most notably in the composition for series in infinitely many variables which has come to be called plethysm. We recall that the plethysm of a series /?(x,, x2,...) with a series u(x,, x2 ,...) is defined as follows. Set c(,(x,, x2 ,...) := a(~,,, x2,, ,... ); the plethysm of p with a is then the series /?(a,, a,,...). There are other applications as well. As is well known, a Lagrange inversion formula requires the understanding of the embedding of the ring of formal power series into a ring of formal Laurent series (which turns out to be a field for series with coefficients in a field). This is an old and thorny problem, which also has not received to this day a satisfactory answer. There are at least two possible definitions of formal Laurent series in many variables [2, 81: they can be required to be either series such that the set of their monomials (which are always of finite length) has an infimum, or series for which the set,of weights of their monomials has a minimum; in this second case it is necessary to add a finiteness condition on monomials