## Algebra & Number Theory

### Modularity of the concave composition generating function

#### Abstract

A composition of an integer constrained to have decreasing then increasing parts is called concave. We prove that the generating function for the number of concave compositions, denoted $v(q)$, is a mixed mock modular form in a more general sense than is typically used.

We relate $v(q)$ to generating functions studied in connection with “Moonshine of the Mathieu group” and the smallest parts of a partition. We demonstrate this connection in four different ways. We use the elliptic and modular properties of Appell sums as well as $q$-series manipulations and holomorphic projection.

As an application of the modularity results, we give an asymptotic expansion for the number of concave compositions of $n$. For comparison, we give an asymptotic expansion for the number of concave compositions of $n$ with strictly decreasing and increasing parts, the generating function of which is related to a false theta function rather than a mock theta function.

#### Article information

Source
Algebra Number Theory, Volume 7, Number 9 (2013), 2103-2139.

Dates
Revised: 10 September 2012
Accepted: 22 October 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730089

Digital Object Identifier
doi:10.2140/ant.2013.7.2103

Mathematical Reviews number (MathSciNet)
MR3152010

Zentralblatt MATH identifier
1282.05016

#### Citation

Andrews, George; Rhoades, Robert; Zwegers, Sander. Modularity of the concave composition generating function. Algebra Number Theory 7 (2013), no. 9, 2103--2139. doi:10.2140/ant.2013.7.2103. https://projecteuclid.org/euclid.ant/1513730089

#### References

• G. E. Andrews, “Ramanujan's “lost” notebook, I: Partial $\theta$-functions”, Adv. in Math. 41:2 (1981), 137–172.
• G. E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, 1998.
• G. E. Andrews, “The number of smallest parts in the partitions of $n$”, J. Reine Angew. Math. 624 (2008), 133–142.
• G. E. Andrews, “Concave compositions”, Electron. J. Combin. 18:2 (2011), Paper 6, 13.
• G. E. Andrews, “Concave and convex compositions”, Ramanujan J. 31:1-2 (2013), 67–82.
• G. E. Andrews, F. G. Garvan, and J. Liang, “Combinatorial interpretations of congruences for the spt-function”, Ramanujan J. 29:1-3 (2012), 321–338.
• W. N. Bailey, “On the basic bilateral hypergeometric series ${}\sb 2\Psi\sb 2$”, Quart. J. Math., Oxford Ser. $(2)$ 1 (1950), 194–198.
• K. Bringmann, “On the explicit construction of higher deformations of partition statistics”, Duke Math. J. 144:2 (2008), 195–233.
• K. Bringmann and A. Folsom, “On the asymptotic behavior of Kac-Wakimoto characters”, Proc. Amer. Math. Soc. 141:5 (2013), 1567–1576.
• K. Bringmann and K. Mahlburg, “An extension of the Hardy–Ramanujan circle method and applications to partitions without sequences”, Amer. J. Math. 133:4 (2011), 1151–1178.
• K. Bringmann and K. Mahlburg, “Asymptotic formulas for coefficients of Kac-Wakimoto characters”, Math. Proc. Cambridge Philos. Soc. 155:1 (2013), 51–72.
• K. Bringmann, K. Mahlburg, and R. C. Rhoades, “Taylor coefficients of mock-Jacobi forms and moments of partition statistics”, preprint, 2012, math.stanford.edu/~rhoades/FILES/RCZ.pdf.
• J. Bryson, K. Ono, S. Pitman, and R. C. Rhoades, “Unimodal sequences and quantum and mock modular forms”, Proc. Natl. Acad. Sci. USA 109:40 (2012), 16063–16067.
• M. C. N. Cheng, “$K3$ surfaces, $N=4$ dyons and the Mathieu group $M\sb {24}$”, Commun. Number Theory Phys. 4:4 (2010), 623–657.
• Y.-S. Choi, “The basic bilateral hypergeometric series and the mock theta functions”, Ramanujan J. 24:3 (2011), 345–386.
• J. Coates, “The work of Gross and Zagier on Heegner points and the derivatives of $L$-series”, pp. 57–72 in Séminaire Bourbaki, $1984/85$, Astérisque 133-134, Société Mathématique de France, Paris, 1986.
• P. Diaconis, S. Janson, and R. C. Rhoades, “Note on a partition limit theorem for rank and crank”, Bull. Lond. Math. Soc. 45:3 (2013), 551–553.
• T. Eguchi and K. Hikami, “Superconformal algebras and mock theta functions, II: Rademacher expansion for $K3$ surface”, Commun. Number Theory Phys. 3:3 (2009), 531–554.
• T. Eguchi, H. Ooguri, and Y. Tachikawa, “Notes on the $K3$ surface and the Mathieu group $M\sb {24}$”, Exp. Math. 20:1 (2011), 91–96.
• N. J. Fine, Basic hypergeometric series and applications, Mathematical Surveys and Monographs 27, American Mathematical Society, Providence, RI, 1988.
• B. Fristedt, “The structure of random partitions of large integers”, Trans. Amer. Math. Soc. 337:2 (1993), 703–735.
• G. Gasper and M. Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, 2004.
• B. H. Gross and D. B. Zagier, “Heegner points and derivatives of $L$-series”, Invent. Math. 84:2 (1986), 225–320.
• S. Heubach and T. Mansour, Combinatorics of compositions and words, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2010.
• R. Lawrence and D. Zagier, “Modular forms and quantum invariants of $3$-manifolds”, Asian J. Math. 3:1 (1999), 93–107.
• J. Lehner, “On automorphic forms of negative dimension”, Illinois J. Math. 8 (1964), 395–407.
• P. A. MacMahon, “Memoir on the theory of compositions of numbers”, London R. S. Phil. A 184 (1893), 835–901.
• A. Malmendier and K. Ono, “Moonshine and Donaldson invariants of ${\mathbb C}P^2$”, preprint, 2012.
• K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics 102, Amer. Math. Soc., Providence, 2004.
• K. Ono, “Mock theta functions, ranks, and Maass forms”, pp. 119–141 in Surveys in number theory, edited by K. Alladi, Dev. Math. 17, Springer, New York, 2008.
• K. Ono, “Unearthing the visions of a master: harmonic Maass forms and number theory”, pp. 347–454 in Current developments in mathematics, 2008, edited by D. Jerison et al., International Press, 2009.
• B. Pittel, “On a likely shape of the random Ferrers diagram”, Adv. in Appl. Math. 18:4 (1997), 432–488.
• R. C. Rhoades, “Strongly unimodal sequences and mixed mock modular forms”, preprint, 2012, math.stanford.edu/~rhoades/FILES/unimodal.pdf.
• R. C. Rhoades, “Families of quasimodular forms and Jacobi forms: the crank statistic for partitions”, Proc. Amer. Math. Soc. 141:1 (2013), 29–39.
• G. Shimura, “On modular forms of half integral weight”, Ann. of Math. $(2)$ 97 (1973), 440–481.
• J. Sturm, “Projections of $C\sp{\infty }$ automorphic forms”, Bull. Amer. Math. Soc. $($N.S.$)$ 2:3 (1980), 435–439.
• A. M. Vershik, “Asymptotic combinatorics and algebraic analysis”, pp. 1384–1394 in Proceedings of the International Congress of Mathematicians (Zürich, 1994), vol. 2, edited by S. D. Chatterji, Birkhäuser, Basel, 1995.
• D. Zagier, “Vassiliev invariants and a strange identity related to the Dedekind eta-function”, Topology 40:5 (2001), 945–960.
• D. Zagier, “Ramanujan's mock theta functions and their applications (after Zwegers and Ono–Bringmann)”, pp. 143–164 (exposé 986) in Séminaire Bourbaki $2007/2008$, Astérisque 326, Société Mathématique de France, Paris, 2009.
• S. Zwegers, Mock theta functions, Ph.D. thesis, Universiteit Utrecht, 2002, igitur-archive.library.uu.nl/dissertations/2003-0127-094324/full.pdf.