A New Partition Identity Coming from Complex Dynamics
George E. Andrews1 and Rodrigo A. Pérez2
1Mathematics Department, The Pennsylvania State University, University Park, PA 16802, USA
andrews@math.psu.edu
2Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
rperez@math.cornell.edu
Annals of Combinatorics 9 (3) p.245-257 September, 2005
AMS Subject Classification: 37F45, 05A19, 11P81
Abstract:
We present a new identity involving compositions (i.e., ordered partitions of natural numbers). The formula has its origin in complex dynamical systems and appears when counting, in the polynomial family , periodic critical orbits with equivalent itineraries. We give two different proofs of the identity; one following the original approach in dynamics and another with purely combinatorial methods.
Keywords: complex dynamics, Mandelbrot set, integer partitions, combinatorial identities

References:

1. G.E. Andrews, The Theory of Partitions, Cambridge University Press, 1984.

2. A. Douady and J.H. Hubbard, Étude dynamique des polynômes complexes I & II, Publ. Math. Orsay, 1984--85.

3. A. Douady and J.H. Hubbard, On the dynamics of polynomial-like maps, Ann. Sci. Éc. Norm. Sup. 18 (1985) 287--343.

4. D. Eberlein, Rational parameter rays of Multibrot sets, Ph. D. Thesis, Technische Universität München, German, 1999. Available at: ftp://ftp.math.sunysb.edu/theses/ thesis99-2/part1.ps.gz.

5. L. Goldberg, Fixed points of polynomial maps, I, Ann. Sci. Éc. Norm. Sup. 25 (1992) 679--685.

6. L. Goldberg and J. Milnor, Fixed points of polynomial maps, II, Ann. Sci. Éc. Norm. Sup. 26 (1992) 51--98.

7. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press, 1979.

8. E. Lau and D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, Institute for Mathematical Sciences, Stony Brook, Preprint #19, 1994.

9. J.W. Milnor, Dynamics in One Complex Variable, Vieweg & Sohn, 1999.

10. J.W. Milnor, Periodic orbits, external rays and the Mandelbrot set: An expository account, In: Asterisque 261 `Geometrie Complexe et Systemes Dynamiques', (2000) pp. 277--333.

11.  I. Niven, H.S. Zuckerman, and H.L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition, Wiley, 1991.

12. A. Poirier, On the realization of fixed point portraits, (An addendum to "Fixed point portraits" by Goldberg and Milnor), Institute for Mathematical Sciences, Stony Brook, Preprint #20, 1991.

13. D. Schleicher, On fibers and local connectivity of Mandelbrot and Multibrot sets, Institute for Mathematical Sciences, Stony Brook, Preprint #13(a), 1998.

1. G.E. Andrews, The Theory of Partitions, Cambridge University Press, 1984.

2. A. Douady and J.H. Hubbard, Étude dynamique des polynômes complexes I & II, Publ. Math. Orsay, 1984--85.

3. A. Douady and J.H. Hubbard, On the dynamics of polynomial-like maps, Ann. Sci. Éc. Norm. Sup. 18 (1985) 287--343.

4. D. Eberlein, Rational parameter rays of Multibrot sets, Ph. D. Thesis, Technische Universität München, German, 1999. Available at: ftp://ftp.math.sunysb.edu/theses/ thesis99-2/part1.ps.gz.

5. L. Goldberg, Fixed points of polynomial maps, I, Ann. Sci. Éc. Norm. Sup. 25 (1992) 679--685.

6. L. Goldberg and J. Milnor, Fixed points of polynomial maps, II, Ann. Sci. Éc. Norm. Sup. 26 (1992) 51--98.

7. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press, 1979.

8. E. Lau and D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, Institute for Mathematical Sciences, Stony Brook, Preprint #19, 1994.

9. J.W. Milnor, Dynamics in One Complex Variable, Vieweg & Sohn, 1999.

10. J.W. Milnor, Periodic orbits, external rays and the Mandelbrot set: An expository account, In: Asterisque 261 `Geometrie Complexe et Systemes Dynamiques', (2000) pp. 277--333.

11. I. Niven, H.S. Zuckerman, and H.L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition, Wiley, 1991.

12. A. Poirier, On the realization of fixed point portraits, (An addendum to "Fixed point portraits" by Goldberg and Milnor), Institute for Mathematical Sciences, Stony Brook, Preprint #20, 1991.

13. D. Schleicher, On fibers and local connectivity of Mandelbrot and Multibrot sets, Institute for Mathematical Sciences, Stony Brook, Preprint #13(a), 1998.


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