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Distribution Properties for t-Hooks in Partitions.
- Source :
-
Annals of Combinatorics . Sep2021, Vol. 25 Issue 3, p677-695. 19p. - Publication Year :
- 2021
-
Abstract
- Partitions, the partition function p(n), and the hook lengths of their Ferrers–Young diagrams are important objects in combinatorics, number theory, and representation theory. For positive integers n and t, we study p t e (n) (resp. p t o (n) ), the number of partitions of n with an even (resp. odd) number of t-hooks. We study the limiting behavior of the ratio p t e (n) / p (n) , which also gives p t o (n) / p (n) , since p t e (n) + p t o (n) = p (n) . For even t, we show that lim n → ∞ p t e (n) p (n) = 1 2 , and for odd t, we establish the non-uniform distribution lim n → ∞ p t e (n) p (n) = 1 2 + 1 2 (t + 1) / 2 if 2 ∣ n , 1 2 - 1 2 (t + 1) / 2 otherwise. Using the Rademacher circle method, we find an exact formula for p t e (n) and p t o (n) , and this exact formula yields these distribution properties for large n. We also show that for sufficiently large n, the sign of p t e (n) - p t o (n) is periodic. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02180006
- Volume :
- 25
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Annals of Combinatorics
- Publication Type :
- Academic Journal
- Accession number :
- 151976300
- Full Text :
- https://doi.org/10.1007/s00026-021-00547-2