Plethystic exponential
In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions, as a concise relation between the generating series for elementary, complete and power sums homogeneous symmetric polynomials in many variables. Its name comes from the operation called plethysm, defined in the context of so-called lambda rings. In combinatorics, the plethystic exponential is a generating function for many well studied sequences of integers, polynomials or power series, such as the number of integer  partitions. It is also an important technique in the enumerative combinatorics of unlabelled graphs, and many other combinatorial objects. In geometry and topology, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariant of its symmetric products. Definition, main properties and basic examples. Let formula_1 be a ring of formal power series in the variable formula_2, with coefficients in a commutative ring formula_3. Denote by be the ideal of power series without constant term. Then, given formula_5 its plethystic exponential, denoted formula_6 is given by where formula_8 is the usual exponential function. It is  readily verified that (writing simply formula_6 when the variable is understood): Some basic examples are: In this last example, formula_12 is number of partitions of formula_13. The plethystic exponential can be also defined for power series rings in many variables. Product-sum formula. The plethystic exponential can be used to provide innumerous product-sum identities. This is a consequence of a product formula for plethystic exponentials themselves. If formula_14 denotes a formal power series with real coefficients formula_15, then it is not difficult to show that:formula_16The analogous product expression also holds in the many variables case. One particularly interesting  case is its relation to integer partitions and to the cycle index of the symmetric group. Relation with symmetric functions. Working with variables formula_17, denote by formula_18 the complete homogeneous symmetric polynomial, that is the sum of all monomials of degree "k" in the variables formula_19, and by formula_20 the elementary symmetric polynomials. Then, the formula_18 and the formula_20 are related to the power sum polynomials: formula_23 by Newton's identities, that can succinctly be written, using plethystic exponentials, as: Macdonald's formula for symmetric products. Let "X" be a finite CW complex, of dimension "d", with Poincaré polynomialformula_26where formula_27  is its "k"th Betti number. Then the Poincaré polynomial of the "n"th symmetric product of "X", denoted formula_28, is obtained from the series expansion:formula_29 The plethystic programme in physics. In a series of articles, a group of theoretical physicists, including Bo Feng, Amihay Hanany and Yang-Hui He, proposed a programme for systematically counting single and multi-trace gauge invariant operators of supersymmetric gauge theories. In the case of quiver gauge theories of D-branes probing Calabi–Yau singularities, this count is codified in the plethystic exponential of the Hilbert series of the singularity.