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A Note on the Equidistribution of 3-Colour Partitions
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(\zeta ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- \zeta {\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0$ and $\zeta$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand.
Keywords: asymptotics, Wright's circle method.
Mots-clés : partitions 
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Joshua Males. A Note on the Equidistribution of 3-Colour Partitions. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a0/

[1] Bringmann K., Craig W., Males J., Ono K., “Distributions on partitions arising from Hilbert schemes and hook lengths”, Forum Math. Sigma, 10 (2022), e49, 30 pp., arXiv: 2109.10394 | DOI | MR | Zbl

[2] Campbell R., Les intégrales eulériennes et leurs applications. Étude approfondie de la fonction gamma, Collection Universitaire de Mathématiques, 20, Dunod, Paris, 1966 | MR

[3] Chern S., “Nonmodular infinite products and a conjecture of Seo and Yee”, Adv. Math., 417 (2023), 108932, 40 pp., arXiv: 1912.10341 | DOI | MR | Zbl

[4] Ciolan A., “Equidistribution and inequalities for partitions into powers”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 66 (2023), 409–431, arXiv: 2002.05682 | MR

[5] Craig W., “Seaweed algebras and the index statistic for partitions”, J. Math. Anal. Appl., 528 (2023), 127544, 23 pp., arXiv: 2112.09269 | DOI | MR

[6] Hardy G.H., Ramanujan S., “Asymptotic formulae in combinatory analysis”, Proc. Lond. Math. Soc., 17 (1918), 75–115 | DOI | MR

[7] Liu Z.-G., Zhou N.H., “Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions”, Ramanujan J., 57 (2022), 1085–1123, arXiv: 1903.00835 | DOI | MR | Zbl

[8] Ngo H.T., Rhoades R.C., “Integer partitions, probabilities and quantum modular forms”, Res. Math. Sci., 4 (2017), 17, 36 pp. | DOI | MR | Zbl

[9] Schlosser M.J., Zhou N.H., Expansions of averaged truncations of basic hypergeometric series, arXiv: 2307.10821

[10] Wright E.M., Stacks, Quart. J. Math. Oxford Ser. (2), 19 (1968), 313–320 | DOI | MR | Zbl

[11] Wright E.M., “Stacks. II”, Quart. J. Math. Oxford Ser. (2), 22 (1971), 107–116 | DOI | MR | Zbl

[12] Zhou N.H., “Note on partitions into polynomials with number of parts in an arithmetic progression”, Int. J. Number Theory, 17 (2021), 1951–1963, arXiv: 1909.13549 | DOI | MR | Zbl