Asymptotics for $k$-crank of $k$-colored partitions

Helen W. J. Zhang; Ying Zhong

Asymptotics for $k$-crank of $k$-colored partitions

Combinatorics 2023-04-14 v1 Number Theory

Authors: Helen W. J. Zhang , Ying Zhong

Abstract

In this paper, we obtain asymptotic formulas for $k$ -crank of $k$ -colored partitions. Let $M_k(a, c; n)$ denote the number of $k$ -colored partitions of $n$ with a $k$ -crank congruent to $a$ mod $c$ . For the cases $k=2,3,4$ , Fu and Tang derived several inequality relations for $M_k(a, c; n)$ using generating functions. We employ the Hardy-Ramanujan Circle Method to extend the results of Fu and Tang. Furthermore, additional inequality relations for $M_k(a, c; n)$ have been established, such as logarithmic concavity and logarithmic subadditivity.

Keywords

k-colored partitions k-crank asymptotic formulas hardy-ramanujan circle method logarithmic concavity generating functions

Cite

@article{arxiv.2304.06316,
  title  = {Asymptotics for $k$-crank of $k$-colored partitions},
  author = {Helen W. J. Zhang and Ying Zhong},
  journal= {arXiv preprint arXiv:2304.06316},
  year   = {2023}
}

Comments

40 pages. arXiv admin note: text overlap with arXiv:1311.4344 by other authors

Asymptotics for $k$-crank of $k$-colored partitions

Abstract

Keywords

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