On the polynomiality and asymptotics of moments of sizes for random $(n, dn\pm 1)$-core partitions with distinct parts
| Autor: | Wenston J.T. Zang, Huan Xiong |
|---|---|
| Přispěvatelé: | Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics (HIT Harbin Institute of Technology), Harbin Institute of Technology (HIT) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Coprime integers
General Mathematics 0102 computer and information sciences 01 natural sciences 05A17 11P81 010101 applied mathematics Combinatorics Average size 010201 computation theory & mathematics [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics Mathematics - Combinatorics Partition (number theory) Combinatorics (math.CO) 0101 mathematics Mathematics |
| Zdroj: | Science in China Series A: Mathematics Science in China Series A: Mathematics, Springer Verlag, In press, ⟨10.1007/s11425-018-9500-x⟩ |
| ISSN: | 1006-9283 1862-2763 |
| DOI: | 10.1007/s11425-018-9500-x⟩ |
| Popis: | Amdeberhan's conjectures on the enumeration, the average size, and the largest size of $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-core partitions with distinct parts, respectively. Let $X_{s,t}$ be the size of a uniform random $(s,t)$-core partition with distinct parts when $s$ and $t$ are coprime to each other. Some explicit formulas for the $k$-th moments $\mathbb{E} [X_{n,n+1}^k]$ and $\mathbb{E} [X_{2n+1,2n+3}^k]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ in this paper, by studying the beta sets of core partitions. In particular, we show that these $k$-th moments are asymptotically some polynomials of n with degrees at most $2k$, when $d$ is given and $n$ tends to infinity. Moreover, when $d=1$, we derive that the $k$-th moment $\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\left(n^2/10\right)^k$ when $n$ tends to infinity. The explicit formulas for the expectations $\mathbb{E} [X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n,dn-1}$. Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics |
| Databáze: | OpenAIRE |
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