In this paper, we examine the functions $a(n)$ and $b(n)$, which respectively represent the number of cubic partitions and cubic partition pairs. Our work leads to the derivation of asymptotic formulas for both $a(n)$ and $b(n)$. Additionally, we establish the upper and lower bounds of these functions, factoring in the explicit error terms involved. Crucially, our findings reveal that $a(n)$ and $b(n)$ both satisfy several inequalities such as log-concavity, third-order Turán inequalities, and strict log-subadditivity.
Chong LI
,
Yi PENG
,
Helen W.J. ZHANG
. INEQUALITIES FOR THE CUBIC PARTITIONS AND CUBIC PARTITION PAIRS[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 737
-754
.
DOI: 10.1007/s10473-025-0225-4
[1] Abramowitz M, Stegun I A.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972
[2] Beckwith O, Bessenrodt C. Multiplicative properties of the number of $k$-regular partitions. Ann Comb, 2016, 20(2): 231-250
[3] Bessenrodt C, Ono K. Maximal multiplicative properties of partitions. Ann Comb, 2016, 20(1): 59-64
[4] Bringmann K, Kane B, Rolen L, Tripp Z. Fractional partitions and conjectures of Chern-Fu-Tang and Heim-Neuhauser. Trans Amer Math Soc Ser B, 2021, 8: 615-634
[5] Chan H C. Ramanujan's cubic continued fraction and an analog of his "most beautiful identity". Int J Number Theory, 2010, 6(4): 673-680
[6] Chan H C. Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function. Int J Number Theory, 2010, 6(4): 819-834
[7] Chan H C. Distribution of a certain partition function modulo powers of primes. Acta Math Sin Engl Ser, 2011, 27(4): 625-634
[8] Chen W Y C. The spt-function of Andrews// Surveys in Combinatorics. Cambridge: Cambridge Univ Press, 2017: 141-203
[9] Chen W Y C, Jia D X Q, Wang L X W. Higher order Turán inequalities for the partition function. Trans Amer Math Soc, 2019, 372: 2143-2165
[10] Dawsey M L, Masri R. Effective bounds for the Andrews spt-function. Forum Math, 2019, 31(3): 743-767
[11] Dimitrov D K. Higher order Turán inequalities. Proc Amer Math Soc, 1998, 126(7): 2033-2037
[12] Desalvo S, Pak I. Log-concavity of the partition function. Ramanujan J, 2015, 38(1): 61-73
[13] Dong J J W, Ji K Q, Jia D X Q. Turán inequalities for the broken $k$-diamond partition function. Ramanujan J, 2023, 62(2): 593-615
[14] Engel B. Log-concavity of the overpartition function. Ramanujan J, 2017, 43(2): 229-241
[15] Griffin M, Ono K, Rolen L, Zagier D. Jensen polynomials for the Riemann zeta function and other sequences. Proc Natl Acad Sci USA, 2019, 116(23): 11103-11110
[16] Jia D X Q. Inequalities for the broken $k$-diamond partition functions. J Number Theory, 2023, 249: 314-347
[17] Kim B. Partition statistics for cubic partition pairs. Electron J Combin, 2011, 18(1): Paper 128
[18] Kim B, Toh P C. On the crank function of cubic partition pairs. Ann Comb, 2018, 22(4): 803-818
[19] Liu E Y S, Zhang H W J. Inequalities for the overpartition function. Ramanujan J, 2021, 54(3): 485-509
[20] Nicolas J L. Sur les entiers $N$ pour lesquels il y a beaucoup de groupes abéliens d'ordre $N$. Ann Inst Fourier, 1978, 28(4): 1-16
[21] Szegö G. On an inequality of P. Turán concerning Legendre polynomials. Bull Amer Math Soc, 1948, 54: 401-405
[22] Sussman E.Rademacher series for $\eta$-quotients. arXiv:1710.03415
[23] Zhao H, Zhong Z. Ramanujan type cogruences for a partition function. Electr J Comb, 2011, 18(1): Paper 58
