Confirming Two Conjectures About the Integer Partitions
✍ Scribed by Boris Pittel
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 127 KB
- Volume
- 88
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
✦ Synopsis
For a given integer n, let 4 n denote the set of all integer partitions * 1 * 2 } } } * m >0 (m 1), of n. For the dominance order ``P'' on 4 n , we show that if two partitions *, + are both chosen from 4 n uniformly at random, and independent of each other, then Pr(*P +) Ä 0 as n Ä . This statement answers affirmatively a question posed by Macdonald in 1979. The proof is based on the limit joint distribution of the largest parts counts found by Fristedt. A slight modification of the argument confirms a conjecture made by Wilf in 1982, namely that, for n even, the probability of a random partition being graphical is zero in the limit. The proof of the latter follows the footsteps of Erdo s and Richmond who saw that to confirm Wilf's conjecture it would be sufficient to show that the probability of the first k Erdo s Gallai conditions of a partition being graphical approaches 0 as n, and then k approach infinity. The reason that the proofs of two seemingly unrelated conjectures turned out to be so close is that, as the E-R analysis revealed, the (joint) distribution of the largest part sizes in a partition * and its dual *$ coincides, in the limit, with the distribution of the largest part sizes for two independent partitions.