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In mathematics, plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions, as the concrete 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.^[1]^[2]

The plethystic exponential can be viewed as an infinite form of Weierstrass elementary factors in the theory of entire functions within the classic field of complex analysis.

In geometry and topology, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariants of its symmetric products.^[3]

Definition, main properties and basic examples

Let $R[[x]]$ be a ring of formal power series in the variable $x$ , with coefficients in a commutative ring $R$ . Denote by

R^{0}[[x]]\subset R[[x]]

be the ideal of power series without constant term. Then, given $f(x)\in R^{0}[[x]]$ its plethystic exponential, denoted $PE[f]$ is given by

PE[f](x)=\exp \left(\sum _{k=1}^{\infty }{\frac {f(x^{k})}{k}}\right)

where $\exp(g(x))$ is the usual exponential function. It is readily verified that (writing simply $PE[f]$ when the variable is understood):

{\begin{aligned}[ll]PE[0]&=1\\PE[f+g]&=PE[f]PE[g]\\PE[-f]&=PE[f]^{-1}\end{aligned}}

Some basic examples are:

{\begin{aligned}[ll]PE[x]&={\frac {1}{1-x}}\\PE[x^{n}]&={\frac {1}{1-x^{n}}}\\PE\left[{\frac {x}{1-x}}\right]&=1+\sum _{n\geq 1}p(n)x^{n}\end{aligned}}

In this last example, $p(n)$ is number of partitions of $n\in \mathbb {N}$ .

The plethystic programme in Mathematical-Physics

In a series of articles, a group of mathematical physicists, including Bo Feng, Amihay Hanany and Yang-Hui He, proposed a programme for systematically counting single and multitrace 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.

References

^ Pólya, G.; Read, R. C. (1987). Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. New York, NY: Springer New York. doi:10.1007/978-1-4612-4664-0. ISBN 978-1-4612-9105-3.
^ Harary, Frank (1955-02-01). "The number of linear, directed, rooted, and connected graphs". Transactions of the American Mathematical Society. 78 (2): 445–445. doi:10.1090/S0002-9947-1955-0068198-2. ISSN 0002-9947.
^ Macdonald, I. G. (1962-10). "The Poincare Polynomial of a Symmetric Product". Mathematical Proceedings of the Cambridge Philosophical Society. 58 (4): 563–568. doi:10.1017/S0305004100040573. ISSN 0305-0041. {{cite journal}}: Check date values in: |date= (help)

Feng, Bo; Hanany, Amihay; He, Yang-Hui (2007), "Counting gauge invariants: the plethystic program", Journal of High Energy Physics, 2007, doi:10.1088/1126-6708/2007/03/090, MR 0010594

Littlewood, D. E. (1944), "Invariant theory, tensors and group characters", Philosophical Transactions of the Royal Society A, 239: 305–365, doi:10.1098/rsta.1944.0001, JSTOR 91389, MR 0010594

[1] Pólya, G.; Read, R. C. (1987). Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. New York, NY: Springer New York. doi:10.1007/978-1-4612-4664-0. ISBN 978-1-4612-9105-3.

[2] Harary, Frank (1955-02-01). "The number of linear, directed, rooted, and connected graphs". Transactions of the American Mathematical Society. 78 (2): 445–445. doi:10.1090/S0002-9947-1955-0068198-2. ISSN 0002-9947.

[3] Macdonald, I. G. (1962-10). "The Poincare Polynomial of a Symmetric Product". Mathematical Proceedings of the Cambridge Philosophical Society. 58 (4): 563–568. doi:10.1017/S0305004100040573. ISSN 0305-0041. {{cite journal}}: Check date values in: |date= (help)

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-In [[mathematics]], '''plethytic exponential''' is a certain [[Operator (mathematics)|operator]] defined on (formal) [[power series]] which, like the usual [[Exponential|exponential]], translates addition into multiplication. It appears naturally in the theory of [[symmetric function]]s, and its name comes from the operation called [[plethysm]], defined in the context of so-called [[lambda ring]]s. Historically, one of it’s first important uses has been in the enumerative combinatorics of unlabelled [[graphs]].
+In [[mathematics]], '''plethystic exponential''' is a certain [[Operator (mathematics)|operator]] defined on (formal) [[power series]] which, like the usual [[Exponential|exponential]], translates addition into multiplication. This exponential operator appears naturally in the theory of [[symmetric function]]s, as the concrete relation between the [[Generating function|generating series]] for [[Elementary symmetric polynomial|elementary]], [[Complete homogeneous symmetric polynomial|complete]] and [[Power sum symmetric polynomial|power sums]] homogeneous symmetric polynomials in many variables. Its name comes from the operation called [[plethysm]], defined in the context of so-called [[lambda ring]]s.
+In [[combinatorics]], the plethystic exponential is a [[generating function]] for many well studied sequences of [[Integer|integers]], [[Polynomial|polynomials]] or power series, such as the number of integer [[List of partition topics|partitions]]. It is also an important technique in the [[enumerative combinatorics]] of unlabelled [[graphs]], and many other combinatorial objects.<ref>{{Cite book|last=Pólya|first=G.|url=http://link.springer.com/10.1007/978-1-4612-4664-0|title=Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds|last2=Read|first2=R. C.|date=1987|publisher=Springer New York|isbn=978-1-4612-9105-3|location=New York, NY|language=en|doi=10.1007/978-1-4612-4664-0}}</ref><ref>{{Cite journal|last=Harary|first=Frank|date=1955-02-01|title=The number of linear, directed, rooted, and connected graphs|url=http://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1955-0068198-2|journal=Transactions of the American Mathematical Society|language=en|volume=78|issue=2|pages=445–445|doi=10.1090/S0002-9947-1955-0068198-2|issn=0002-9947}}</ref>
-In [[complex analysis]], the plethystic exponential is related to [[Weierstrass product|Weierstrass product expansions]] of [[Entire function|entire functions]].
+The plethystic exponential can be viewed as an infinite form of [[Weierstrass factorization theorem|Weierstrass elementary factors]] in the theory of [[Entire function|entire functions]] within the classic field of [[complex analysis]].
-In [[combinatorics]], the plethystic exponential is a [[generating function]] for many well studied sequences of [[Integer|integers]], [[Polynomial|polynomials]] or power series.
-In [[geometry]] and [[topology]], the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariants of its symmetric products.
+In [[geometry]] and [[topology]], the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariants of its symmetric products.<ref>{{Cite journal|last=Macdonald|first=I. G.|date=1962-10|title=The Poincare Polynomial of a Symmetric Product|url=https://www.cambridge.org/core/product/identifier/S0305004100040573/type/journal_article|journal=Mathematical Proceedings of the Cambridge Philosophical Society|language=en|volume=58|issue=4|pages=563–568|doi=10.1017/S0305004100040573|issn=0305-0041}}</ref>
 == Definition, main properties and basic examples==