Recursive matrices and umbral calculus

Abstract

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