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Confirming two conjectures about the integer partitions. (English) Zbl 0937.05008
The author answers a question posed by I. G. Macdonald in [Symmetric functions and Hall polynomials. 2nd ed. (Clarendon Press, Oxford) (1995; Zbl 0824.05059)]. He proves that if two partitions, $$\lambda$$ and $$\mu$$, are chosen uniformly at random and independent of each other from the set of partitions of $$n$$, then the probability that $$\lambda$$ and $$\mu$$ are comparable in the usual dominance order approaches 0 as $$n$$ approaches infinity. From the Gale-Ryser theorem, it follows that if $$\pi_n$$ is the probability that there exists a bipartite graph on $$(X,Y)$$ such that $$\lambda$$ and $$\mu$$ are the degree sequences of the respective vertex sets, then $$\lim_{n\to \infty} \pi_n = 0$$. The same methods enable the author to prove a conjecture made by Wilf in 1982: The probability that a randomly chosen partition of $$n$$ is the degree sequence of a graph approaches 0 as $$n$$ approaches infinity.

##### MSC:
 05A17 Combinatorial aspects of partitions of integers 05C07 Vertex degrees 60F05 Central limit and other weak theorems 60F20 Zero-one laws 60C05 Combinatorial probability
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##### References:
 [1] Auluck, F.C.; Chowla, S.; Gupta, H., On the maximum value of partitions of n into k parts, J. Indian math. soc., 6, 105-112, (1942) · Zbl 0063.00139 [2] Barnes, T.M.; Savage, C.D., A recurrence for counting graphical partitions, Electron. J. combin., 2, (1995) · Zbl 0818.05003 [3] Brualdi, R.A.; Ryser, H.J., Combinatorial matrix theory, Encyclopedia of mathematics and its applications, (1992), Cambridge University Press Cambridge [4] Brylawski, T., The lattice of integer partitions, Discrete mathematics, 6, 201-219, (1973) · Zbl 0283.06003 [5] Diaconis, P., Group representations in probability and statistics, Institute of mathematical statistics, Lecture notesâ€”monograph series, 11, (1988), Hayward · Zbl 0695.60012 [6] Erdős, P.; Gallai, T., Graphs with given degree of vertices, Mat. lapok, 11, 264-274, (1960) [7] Erdős, P.; Lehner, J., The distribution of the number of summands in the partition of a positive integer, Duke math. J., 8, 335-345, (1941) · JFM 67.0126.02 [8] Erdős, P.; Richmond, L.B., On graphical partitions, Combinatorica, 13, 57-63, (1993) [9] Feller, W., An introduction to probability theory and its applications, II, (1971), John Wiley & Sons New York · Zbl 0219.60003 [10] Fristedt, B., The structure of random partitions of large integers, Trans. amer. math. soc., 337, 703-735, (1993) · Zbl 0795.05009 [11] Macdonald, I.G., Symmetric functions and Hall polynomials, 2nd edition, (1995), Oxford Univ. PressOxford Sci. Publ London · Zbl 0487.20007 [12] Pittel, B., On a likely shape of the random Ferrers diagram, Adv. in appl. math., 18, 432-488, (1997) · Zbl 0894.11039 [13] Rousseau, C.; Ali, F., On a conjecture concerning graphical partitions, Congr. numer., 104, 150-160, (1994) · Zbl 0836.05006 [14] Sagan, B.E., The symmetric group: representations, combinatorial algorithms, & symmetric functions, (1991), Wadsworth & Brooks/Cole Pacific Grove · Zbl 0823.05061 [15] Sierksma, G.; Hoogeveen, H., Seven criteria for integer sequences being graphic, Journal of graph theory, 15, 223-231, (1991) · Zbl 0752.05052
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