A note on Abel polynomials and rooted labeled forests
A special case of the Abel polynomials counts rooted labeled forests. This interpretation is used to obtain a combinatorial proof of the formula expressing x^n as a sum of these polynomials.
On the (co)homology of the poset of weighted partitions
We consider the poset of weighted partitions $\Pi_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\Pi_n^w$ provide a generalization of the lattice $\Pi_n$ of partitions, which we show possesses many of the well-known properties of $\Pi_n$. In particular, we prove these intervals are EL-shellable, we show that the Mobius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\{1,2,\dots,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $\mathfrak{S}_n$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $\Pi_n^w$ has a nice factorization analogous to that of $\Pi_n$.
A new expression for umbral operators and power series inversion
It is shown here the compositional inversion of formal power series has a natural formulation in terms of umbral operators. This fact is used to derive a new proof of the Lagrange inversion theorem by operator theoretic methods. Introduction. The main object of this paper is to show that the theory of umbral operators as developed in [1] offers a natural setting for the study of formal power series inversion. A by-product of this idea is perhaps one of the most elementary approaches to the Lagrange inversion theorem in any number of variables. However, we shall restrict ourselves here to the one-dimensional case leaving the higher-dimensional formulas, as well as some of the applications, to a forthcoming publication. This paper is divided into two sections. In the first section we recall some of the basic ingredients of the theory of polynomials of binomial type as developed in [1]. In the second section we obtain a new expression for umbral operators which is, in a sense, "dual" to the Lagrange inversion formula. From this expression, several forms of Lagrange's theorem can be immediately derived. We are indebted to G.-C. Rota for stimulating conversations and encouragement. Indeed, it was during one of our endless discussions with him that the idea developed to put the umbral calculus to such a use. 1. Basics on the theory of umbral operators. Our notation follows very closely that adopted in [ 1 ] and the reader would do well to read that paper for a more complete background. We shall deal with polynomial operators, namely linear maps of polynomials into polynomials. Here, we shall work with polynomials in ^[x], that is, polynomials in the variable x with coefficients in a given field ?F of characteristic zero. For sake of simplicity the reader may restrict f to be the real number field. This given, we shall express all our formulas by means of the following basic operators: (1) L0 = Evaluation at 0: Received by the editors November 23, 1976. AMS (MOS) subject classifications (1970). Primary 05A15: Secondary 47E05. © American Mathematical Society 1977 179 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 180 A. M. GARSIA AND S. A. JONI n if »-a 0 \0 if n > 1. (2) D = Differentiation: Dx" = nx"~x, (3) x — Multiplication by x: xx" = x"+i. Given a formal power series $(h) = 2^°=o %u* w^tn coefficients in % we let $(£>) denote the polynomial operator HD) = 5 <¡>„D\ r-0 That is, (1.1) H>(D)xn = 2 *,«(« 1) •••(»r + \)xn-". It can be shown (see [1]) that a polynomial operator Q is of form (1.1) if and only if Q commutes with D. One of the basic tools in [1] as well as here is the fundamental fact that two polynomial operators P and Q that commute with D are identical if and only if (1.2) L0P = L0Q. This follows from the fact that in this case (1.2) implies L0e*DP = L^eyDQ Vy G % from which the equality of P and Q immediately follows. There are two further operations that play a crucial role. These are the two "Lie" derivatives □ and V which act on a polynomial operator P as follows: (a) D P = Px xP, (1.3) (b) V£ = DP PD. The first of these was extensively used in [1] and [2]. Both □ and V are differentiations and we have (a) Do" = nDn~\ (1.4) (b)Vxn = nx"-\ Note that if $(m) is a formal power series then D HD) = $'(£>). We shall make extensive use of the trivial identity (1.5) L0 D P = L0Px, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UMBRAL OPERATORS 181 valid for every polynomial operator P. This was used in [1] in order to express the "Rodriguez" formula of [2] as an identity for umbral operators.1 The essential fact expressed by (1.5) is that the derivative D is the "adjoint" of the operator x with respect to the "scalar product" (P, Q) = L0P(D)Q(x). As in [2] a polynomial operator (1-6) /CD) = I f„D", /, * 0, v= 1 will be called a A-operator. It is easy to see that a polynomial operator Q is of form (1.6) if and only if Q commutes with D,Q\ =0 and Qx ^ 0. Given a A-operator/(Z)) we call (1.7) Uf=ï ^L0f(D) v=0 v' the "umbral operator" associated to /(£>). The sequence of polynomials y„(x) = UtXn will be called the B.T. sequence associated to Uj. Note that since yn(x) = f"x" + • • •, the sequence {yn} is a basis and thus U, is invertible. We also note that we have the basic identity (1.8) DUf = Uff(D). Taking all this into account it is easy to see that an operator U is of form (1.7) if and only if (1) U is invertible, (1.9) (2)L0<7 = L0, (3) U~lDU is a A-operator. Indeed, if U~lDU = f(D) then (1.8) holds and the identity CO » u= 2nfrVr(J» follows immediately upon multiplying Taylor's formula /= sfrV on the right by U. The relevance of all this material to power series inversion comes from the following fundamental fact pointed out in [2], namely: The product U = UjU of two umbral operators is equal to the umbral operator associated to the formal power series See the proof of Theorem 3.3 in [1]. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 182 AM. GARSIA AND S. A. JONI oo /(*(«)) = 2 /,(*(«))', "functional composition" off and g. To see this note that (1.8) implies that for every formal power series <I>(w) we have (1.10) HD)Uf = Uf$(f(D)). This given, we first observe that U = UfUg clearly satisfies both (1.9)(1) and (2). Furthermore, the identities DUf = Uff(D) and DUg = Ugg(D) (combined with (1.10)) yield DUfUg = UfUgf(g(D)). It is well known and easy to show that given any formal power series f(u) — 2^=iX"r(/i ^ 0)> there exists a unique formal power series F(u) = 2r=i py (F\ * °) such that f(F(u)) F(/(")) = "• The existence of F(u) follows immediately from the above considerations since the operator Uj~x can be easily shown to satisfy the conditions in (1.9). But then if Ufx = UF, the fact that UFUt = I immediately yields that F(u) and/(w) are inverses of each other under functional composition. It is to be noted that in this case (1.10) with $(D) replaced by F(D) gives (1.11) F(D)Uf=UfD, and this in turn is equivalent to (1.8). 2. The Lagrange inversion formula. In this section the formal power series F(u) = 2^=1 Fi>u" (Fi ^ 0) will be considered as given, and f(u) = 2^=1 ÎVDV, the "functional inverse" of F(u), will be considered as unknown. The Lagrange inversion formula expresses the coefficients of f(u) directly in terms of the coefficients of F(u). We shall see that in this context this corresponds to obtaining an expression for the umbral operator U, = Up directly in terms of the operator F(D). Since there is no loss of generality we shall write our formal power series F(u) in the form (2.1) F(u) = u G(u) where 00 (2.2) G(u) = 2 Gy. v = l Our basic result can be stated as follows: Theorem 2.1. The inverse of the operator UF is given by License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UMBRAL OPERATORS lgg (2.3) U = F'(D) 5 G*(Z>)-" »■=0 Proof. In view of the above considerations, to show this we need only check that (2.4) F(D)U = UD and (2.5) LQU = Lousing (2.1), (2.4) can be written in the form DU UD = G(D)U, that is, (2.6) VU=G(D)U. However, this follows immediately from (1.4)(b). Indeed, since operators which commute with D behave like constants with respect to V, we ¿et Vc/ = F(¿>)Í^W-\ v=\ v • from which (2.6) immediately follows. As for (2.5) we observe that, in view of the basic identity (1.5), it can be written in the form L0 i D'(1-DCÍ = ¿0, c=0 v • and this immediately follows from the fact that 2 d'(i-dcÇ= 2 \uv~-uv+x7^~^\ = i. x=o v„=o I "' (v + 1)! J The following are two important consequences of the above result. Corollary 2.1. 1^-3 G"(Z>) „_, (2-7) t//1 = / + x 2 Proof. We can write the operator U in (2.3) in the form ,=o d „fo (r + 1)! = y £Í2L»_ ? G'+\D) +x « G,+l(/?) , from which (2.7) follows immediately. Corollary 2.1. /y License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 184 A. M. GARSIA AND S. A. JONI (2.8) f(u) = u + G(f(u)) where G(u) = 2^=2 ^M^> then for any formal power series 4>(w) we have (2.9) $(/(„)) = $(M) + i ¿(¿y [*'(«)<?'(«)]• Proof. Clearly (2.8) says that f(u) is the functional inverse of F(u) = u — G(u). Thus, using (1.8) we get L0 0(/(Z>)) = LqU^HD)) = L0$>(D)Uj. Substituting for ÍA the expression in (2.7) we obtain £0$(/(Z))) = LJ>{D) + 2 £o^(ö)í-^-^í''-, V= 1 = £„$(/?) + 2 LonD)q^x"-x v=l v • = L0[o>(Z))+ 2i¿D"-1{$'G"(D)}]. Our remark about operators commuting with D then yields $>(/(/>)) = $(£>) + 2 -Jr D"-1 {<&'G'(Z>)}, »•=1 '• and this is clearly equivalent to (2.9). We shall conclude by deriving one of the standard forms of the Lagrange inversion formula. Theorem 2.2. Let f(u) be the formal inverse of the formal power series f(u) = 2 py (£, # 0). v=\ Then for any formal power series $(«) we have (2.10) *(/(«))L» = (\/n)V(ú)(u/F(u))n\u.-¿ Proof. Again we may assume with loss that £, = 1 and set (2.11) F(u) = u uH(u). We shall make use of formula (2.7) with G(u) = uH(u). To this end note that *(/(«))L. = (\/n\)L0Hf(D))xn = (l/n\)L0UfHf(D))xn = (l/n\)L0<S>(D)Ufx". 2 Here |„„ means "the coefficient of «"". License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UMBRAL OPERATORS 185 n-\ Substituting (2.7) for U¡ we obtain *(/(«))!„. = ¿r ¿o*P)[*" + í 2, ̂ W+'-'] = ¿^•wf**-1 + i gy(i»w(w+1)-;!(/"h,,~1)x"-'] i *'(«)-n • QEDn K)(\H(u))" References 1. A. M. Garsia, An exposé of the Mullin-Rota theory of polynomials of binomial type, Linear and Multilinear Algebra 1 (1973), 47-65. MR 47 #8321a. 2. R. Mullin and G.-C. Rota, On the theory of binomial enumeration, Graph Theory and Applications, Academic Press, New York, 1970. Department of Mathematics, University of California at San Diego, La Jolla, California 92093 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use