Abstract
We study the distribution of the number of parts of given multiplicity (or equivalently, ascents of given size) in integer partitions. In this paper we give methods to compute asymptotic formulas for the expected value and variance of the number of parts of multiplicity d (d is a positive integer) in a random partition of a large integer n and also prove that the limiting distribution is asymptotically normal for fixed d. However, if we let d increase with n, we get a phase transition for d around n 1/4. Our methods can also be applied to the so−called λ-partitions where the parts are members of a sequence of integers λ.
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References
Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)
Brennan C., Knopfmacher A., Wagner S.: The distribution of ascents of size d or more in partitions of n. Combin. Probab. Comput. 17(4), 495–509 (2008)
Corteel S., Pittel B., Savage C.D., Wilf H.S.: On the multiplicity of parts in a random partition. Random Structures Algorithms 14(2), 185–197 (1999)
Curtiss J.H.: A note on the theory of moment generating functions. Ann. Math. Statistics 13, 430–433 (1942)
Erdös P., Lehner J.: The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, 335–345 (1941)
Flajolet P., Gourdon X., Dumas P.: Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144(1-2), 3–58 (1995)
Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Goh W.M.Y., Schmutz E.: The number of distinct part sizes in a random integer partition. J. Combin. Theory Ser. A 69(1), 149–158 (1995)
Grabner, P.J., Knopfmacher, A., Wagner, S.: A general asymptotic scheme for moments of partition statistics. Preprint
Hwang H.-K.: Limit theorems for the number of summands in integer partitions. J. Combin. Theory Ser. A 96(1), 89–126 (2001)
Knopfmacher A., Munagi A.O.: Successions in integer partitions. Ramanujan J. 18(3), 239–255 (2009)
Madritsch M., Wagner S.: A central limit theorem for integer partitions. Monatsh. Math. 161(1), 85–114 (2010)
Ralaivaosaona D.: On the number of summands in a random prime partition. Monatsh. Math. 166(3), 505–524 (2012)
Roth, K.F., Szekeres, G.: Some asymptotic formulae in the theory of partitions. Quart. J. Math. Oxford Ser. (2) 5, 241–259 (1954)
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Ralaivaosaona, D. On the Distribution of Multiplicities in Integer Partitions. Ann. Comb. 16, 871–889 (2012). https://doi.org/10.1007/s00026-012-0165-2
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DOI: https://doi.org/10.1007/s00026-012-0165-2