Moments of partitions and derivatives of higher order
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Moments of partitions and derivatives of higher order Shaul Zemel1 Received: 26 February 2020 / Accepted: 9 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We define moments of partitions of integers, and show that they appear in higher-order derivatives of certain combinations of functions. Keywords Higher derivatives · Partitions · Moments
1 Introduction and statement of the main result Changes of coordinates grew, through the history of mathematics, from a powerful computational tool to the underlying object behind the modern definition of many objects in various branches of mathematics, like differentiable manifolds or Riemann surfaces. With the change of coordinates, all the objects that depend on these coordinates change their form, and one would like to investigate their behavior. For functions of one variable, like holomorphic functions on Riemann surfaces, this is very easy, but one may ask what happens to the derivatives of functions under this operation. The answer is described by the well-known formula of Faà di Bruno for the derivative of any order of a composite function. For the history of this formula, as well as a discussion of the relevant references, see [3]. For phrasing Faà di Bruno’s formula, we recall that a partition λ of some integer n, denoted by λ n, is defined to be a finite sequence of positive integers, say al with 1 ≤ l ≤ L, written in decreasing order, whose sum is n. The number L is called the length of λ and is denoted by (λ), and given a partition λ, the number n for which λ n is denoted by |λ|. Another method for representing partitions, which will be more useful for our purposes, is by the multiplicities m i with i ≥ 1, which are defined by m i = {1 ≤ l ≤ L|al = i}, with m i ≥ 0 for every i ≥ 1 and such that only finitely many multiplicities are nonzero. In this case, we have |λ| = i≥1 im i and
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Shaul Zemel [email protected] Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmund Safra Campus, 91904 Jerusalem, Israel
123
Journal of Algebraic Combinatorics
(λ) = i≥1 m i . Note that the empty partition, in which all the multiplicities m i vanish, is allowed. It is considered to be partition of 0, with length 0. Therefore, when some partition λ is known from the context, the numbers m i will denote the associated multiplicities, and in case several partitions are involved we may write m i (λ) for clarification. Assume that f is a function of z and the variable z is a function of another variable t, say z = ϕ(t), and we wish to differentiate the resulting function of t successively. The formula of Faà di Bruno is the answer to this question, which we can write explicitly as dn ( f ◦ ϕ)(n) (t) = n ( f (ϕ(t) dt n (i) m i n! ((λ)) n = f ϕ(t) ϕ (t) . (1) mi m ! (i!) i i=1 λn
i=1
We remark that gathering these formulae for all n together, and noticing that λ appears in the derivative of order |λ|, yields a structure of a Hopf algebra on the polynomial ring of infinitely m
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