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Quasipolynomials and Maximal Coefficients of Gaussian Polynomials

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Abstract

We establish an algorithm for producing formulas for p(n, m, N), the function enumerating partitions of n into at most m parts with no part larger than N. Recent combinatorial results of H. Hahn et al. on a collection of partition identities for p(n, 3, N) are considered. We offer direct proofs of these identities and then place them in a larger context of the unimodality of Gaussian polynomials \(N+m\brack m\) whose coefficients are precisely p(n, m, N). We give complete characterizations of the maximal coefficients of \({M\brack 3}\) and \({M\brack 4}\). Furthermore, we prove a general theorem on the period of quasipolynomials for central/maximal coefficients of Gaussian polynomials. We place some of Hahn’s identities into the context of some known results on differences of partitions into at most m parts, p(n, m), which we then extend to p(n, m, N).

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References

  1. Almkvist, G., Andrews, G.E.: A Hardy-Ramanujan formula for restricted partitions. J. Number Theory 38(2), 135–144 (1991)

    Article  MathSciNet  Google Scholar 

  2. Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  3. Andrews, G.E.: Partitions: At the interface of \(q\)-series and modular forms. Rankin memorial issues. Ramanujan J. 7(1-3), 385–400 (2003)

    Article  MathSciNet  Google Scholar 

  4. Barnabei, M., Bonetti, F., Silimbani, M.: Bijections and recurrences for integer partitions into a bounded number of parts. Appl. Math. Lett. 22(3), 297–303 (2009)

    Article  MathSciNet  Google Scholar 

  5. Beck, M., Robins, S.: Computing the Continuous Discretely. Springer, New York (2007)

    MATH  Google Scholar 

  6. Breuer, F.: An invitation to Ehrhart theory: Polyhedral geometry and its applications in enumerative combinatorics. In: Gutierrez, J., Schicho, J., Weimann, M. (eds.) Computer Algebra and Polynomials, Lecture Notes in Computer Science, Vol. 8942, pp. 1–29. Springer, Cham, Switzerland (2015)

    Google Scholar 

  7. Breuer, F., Eichhorn, D., Kronholm, B.: Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts. European J. Combin. 65, 230–252 (2017)

    Article  MathSciNet  Google Scholar 

  8. Dhand, V.: A combinatorial proof of strict unimodality for \(q\)-binomial coefficients. Discrete Math. 335, 20–24 (2014)

    Article  MathSciNet  Google Scholar 

  9. Ehrhart, E.: Sur les polyèdres homothétiques bordés à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 988–990 (1962)

    MathSciNet  MATH  Google Scholar 

  10. Ehrhart, E.: Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux. J. Reine Angew. Math. 226, 1–29 (1967)

  11. Ehrhart, E.: Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires. J. Reine Angew. Math. 227, 25–49 (1967)

  12. Glaisher, J.W.L.: On the number of partitions of a number of partitions of a number into a given number of parts. Quart. J. Pure and Appl. Math. 40, 57–143 (1908)

    MATH  Google Scholar 

  13. Glaisher, J.W.L.: Formulae for partitions into given elements, derived from Sylvester’s theorem. Quart. J. Pure and Appl. Math. 40, 275–348 (1909)

    MATH  Google Scholar 

  14. Glösel, K.: Über die Zerlegung der ganzen Zahlen. Monatsh. Math. Phys. 7(1), 133–141 (1896)

    Article  MathSciNet  Google Scholar 

  15. Gupta, H., Gwyther, C.E., Miller, J.C.P.: Tables of Partitions. Royal Society Math. Tables, Vol. 4, Cambridge University Press, Cambridge (1958)

  16. Hahn, H.: On classical groups detected by the triple tensor products and the Littlewood-Richardson semigroup. Res. Number Theory 2, Art. 19 (2016)

  17. Hahn, H.: On tensor third \(L\)-functions of automorphic representations of \({\rm GL}_n(\mathbb{A}_F)\). Proc. Amer. Math. Soc. 144(12), 5061–5069 (2016)

    Article  MathSciNet  Google Scholar 

  18. Hahn, H., Huh, J., Lim, E., Sohn, J.: From partition identities to a combinatorial approach to explicit Satake inversion. Ann. Comb. 22(3), 543–562 (2018)

    Article  MathSciNet  Google Scholar 

  19. Hardy, G.H., Ramanujan, S.: Asymptotic formulæ in combinatory analysis. Proc. Lond. Math. Soc. 17(2), 75–115 (1918)

    Article  Google Scholar 

  20. Larsen, A.: First differences of \(p(n, m)\). Masters Thesis. University of Texas Rio Grande Valley (2019)

  21. Munagi, A.O.: Computation of \(q\)-partial fractions. Integers 7, #A25 (2007)

  22. O’Hara, K.M.: Unimodality of Gaussian coefficients: a constructive proof. J. Combin. Theory Ser. A 53(1), 29–52 (1990)

    Article  MathSciNet  Google Scholar 

  23. Pak, I., Panova, G.: Strict unimodality of \(q\)-binomial coefficients. C. R. Math. Acad. Sci. Paris 351(11-12), 415–418 (2013)

    Article  MathSciNet  Google Scholar 

  24. Rubinstein, B.Y., Fel, L.G.: Restricted partition functions as Bernoulli and Eulerian polynomials of higher order. Ramanujan J. 11(3), 331–347 (2006)

    Article  MathSciNet  Google Scholar 

  25. Sills, A.V., Zeilberger, D.: Formulæ for the number of partitions of \(n\) into at most \(m\) parts (using the quasi-polynomial ansatz). Adv. in Appl. Math. 48(5), 640–645 (2012)

    Article  MathSciNet  Google Scholar 

  26. Stanley, R.P., Zanello, F.: Some asymptotic results on \(q\)-binomial coefficients. Ann. Comb. 20(3), 623–634 (2016)

    Article  MathSciNet  Google Scholar 

  27. Sylvester, J.J.: Proof of the hitherto undemonstrated fundamental theorem of invariants. In: Collect. Math. Papers 3, Chelsea, New York, 117–126 (1973)

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Acknowledgements

This material is based upon work supported by, or in part by, the NSF-Louis Stokes Alliance for Minority Participation program under grant number HRD-1202008. The authors would like to thank the referee for providing helpful comments, especially with some insight regarding (4.26) and (4.27). The authors are especially grateful for the input and advice of both Drew Sills and Dennis Eichhorn regarding the revisions of this paper. The authors offer hearty thanks to George E. Andrews for alerting us to the recent results of Hahn in [18]. In forthcoming work, we will set

\(\displaystyle \sum _{k=0}^{\infty }{k+m-1 \atopwithdelims ()m-1}q^{\text {lcm}(m) k}=A_m(q)\)

and consider generalizations to p(n, m, N) and \({N+m\brack m}\) by further inquiry into

\(\displaystyle {N+m \brack m}= G_m(q)E_{m}(q)A_m(q). \;\qquad \qquad \qquad \qquad (6.7)\)

It is clear that GEA in (6.7) is a great mathematical object rich with wisdom and generosity and worthy of our attention and admiration.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, 7839-2999, USA

    Angelica Castillo, Stephanie Flores, Brandt Kronholm, Acadia Larsen & Arturo Martinez

  2. College of Engineering and Computer Science, University of Texas Rio Grande Valley, Edinburg, TX, 7839-2999, USA

    Anabel Hernandez

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  1. Angelica Castillo
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  2. Stephanie Flores
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  3. Anabel Hernandez
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  4. Brandt Kronholm
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Correspondence to Brandt Kronholm.

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Dedicated to George Andrews on the occasion of his 80th birthday

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A. Hernandez was partially supported by the US National Science Foundation DUE-1458830.

Appendix A. The 36 Constituents of the Quasipolynomial for p(n, 3, N) Arranged by \(N\pmod 6\)

Appendix A. The 36 Constituents of the Quasipolynomial for p(n, 3, N) Arranged by \(N\pmod 6\)

$$\begin{aligned}&N=6j \nonumber \\ p(6k,3,6j)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2} + \genfrac(){0.0pt}1{k }{ 2} -12\genfrac(){0.0pt}1{k+1-j }{ 2} -6\genfrac(){0.0pt}1{k-j }{ 2} + 6\genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 12\genfrac(){0.0pt}1{k-2j }{ 2} -\genfrac(){0.0pt}1{k+1-3j }{ 2} -4\genfrac(){0.0pt}1{k-3j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.1)
$$\begin{aligned} p(6k+1,3,6j)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 5\genfrac(){0.0pt}1{k+1 }{ 2} -\genfrac(){0.0pt}1{k+2-j }{ 2} -13\genfrac(){0.0pt}1{k+1-j }{ 2} -4\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad +9\genfrac(){0.0pt}1{k+1-2j }{ 2} +9\genfrac(){0.0pt}1{k-2j }{ 2} -\genfrac(){0.0pt}1{k+1-3j }{ 2} -5\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.2)
$$\begin{aligned} p(6k+2,3,6j)&= 2\genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2} - 2\genfrac(){0.0pt}1{k+2-j }{ 2}-14\genfrac(){0.0pt}1{k+1-j }{ 2} -2\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad + 12\genfrac(){0.0pt}1{k+1-2j }{ 2} +6\genfrac(){0.0pt}1{k-2j }{ 2} -2\genfrac(){0.0pt}1{k+1-3j }{ 2} -4\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.3)
$$\begin{aligned} p(6k+3,3,6j)&=3\genfrac(){0.0pt}1{k+2 }{ 2} + 3\genfrac(){0.0pt}1{k+1 }{ 2} - 4\genfrac(){0.0pt}1{k+2-j }{ 2}-13\genfrac(){0.0pt}1{k+1-j }{ 2} -\genfrac(){0.0pt}1{k-j }{ 2} + \genfrac(){0.0pt}1{k+2-2j }{ 2}\nonumber \\&\quad + 13\genfrac(){0.0pt}1{k+1-2j }{ 2}+4\genfrac(){0.0pt}1{k-2j }{ 2} -3\genfrac(){0.0pt}1{k+1-3j }{ 2} -3\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.4)
$$\begin{aligned} p(6k+4,3,6j)&=4\genfrac(){0.0pt}1{k+2 }{ 2} + 2\genfrac(){0.0pt}1{k+1 }{ 2} - 6\genfrac(){0.0pt}1{k+2-j}{ 2}-12\genfrac(){0.0pt}1{k+1-j }{ 2} + 2\genfrac(){0.0pt}1{k+2-2j }{ 2} \nonumber \\&\quad + 14\genfrac(){0.0pt}1{k+1-2j }{ 2} + 2\genfrac(){0.0pt}1{k-2j }{ 2} -4\genfrac(){0.0pt}1{k+1-3j }{ 2} -2\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.5)
$$\begin{aligned} p(6k+5,3,6j)&= 5\genfrac(){0.0pt}1{k+2 }{ 2} + \genfrac(){0.0pt}1{k+1 }{ 2} - 9\genfrac(){0.0pt}1{k+2-j }{ 2}-9\genfrac(){0.0pt}1{k+1-j }{ 2} + 4\genfrac(){0.0pt}1{k+2-2j }{ 2} \nonumber \\&\quad + 13\genfrac(){0.0pt}1{k+1-2j }{ 2}+\genfrac(){0.0pt}1{k-2j }{ 2} -5\genfrac(){0.0pt}1{k+1-3j }{ 2} -\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.6)
$$\begin{aligned}&N=6j+1 \nonumber \\ p(6k,3,6j+1)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}+\genfrac(){0.0pt}1{k }{ 2} - 9\genfrac(){0.0pt}1{k+1-j }{ 2} -9\genfrac(){0.0pt}1{k-j }{ 2} + 2\genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 14\genfrac(){0.0pt}1{k-2j }{ 2} +2\genfrac(){0.0pt}1{k-1-2j }{ 2} -3\genfrac(){0.0pt}1{k-3j }{ 2} -3\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.7)
$$\begin{aligned} p(6k+1,3,6j+1)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 5\genfrac(){0.0pt}1{k+1 }{ 2} - 12\genfrac(){0.0pt}1{k+1-j }{ 2} -6\genfrac(){0.0pt}1{k-j }{ 2} + 4\genfrac(){0.0pt}1{k+1-2j }{ 2}\nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-2j }{ 2} +\genfrac(){0.0pt}1{k-1-2j }{ 2} -4\genfrac(){0.0pt}1{k-3j }{ 2} -2\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.8)
$$\begin{aligned} p(6k+2,3,6j+1)&= 2\genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2} - \genfrac(){0.0pt}1{k+2-j }{ 2} -13\genfrac(){0.0pt}1{k+1-j }{ 2} -4\genfrac(){0.0pt}1{k-j }{ 2}\nonumber \\&\quad + 6\genfrac(){0.0pt}1{k+1-2j }{ 2} + 12\genfrac(){0.0pt}1{k-2j }{ 2} -5\genfrac(){0.0pt}1{k-3j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.9)
$$\begin{aligned} p(6k+3,3,6j+1)&= 3\genfrac(){0.0pt}1{k+2 }{ 2} + 3\genfrac(){0.0pt}1{k+1 }{ 2} - 2\genfrac(){0.0pt}1{k+2-j }{ 2}-14\genfrac(){0.0pt}1{k+1-j }{ 2} -2\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad + 9\genfrac(){0.0pt}1{k+1-2j }{ 2}+ 9\genfrac(){0.0pt}1{k-2j }{ 2} -\genfrac(){0.0pt}1{k+1-3j }{ 2} -4\genfrac(){0.0pt}1{k-3j }{ 2}-\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.10)
$$\begin{aligned} p(6k+4,3,6j+1)&= 4\genfrac(){0.0pt}1{k+2 }{ 2} + 2\genfrac(){0.0pt}1{k+1 }{ 2} - 4\genfrac(){0.0pt}1{k+2-j }{ 2} -13\genfrac(){0.0pt}1{k+1-j }{ 2} -\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad + 12\genfrac(){0.0pt}1{k+1-2j }{ 2} + 6\genfrac(){0.0pt}1{k-2j }{ 2} -\genfrac(){0.0pt}1{k+1-3j }{ 2} -5\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.11)
$$\begin{aligned} p(6k+5,3,6j+1)&= 5\genfrac(){0.0pt}1{k+2 }{ 2} + \genfrac(){0.0pt}1{k+1 }{ 2} - 6\genfrac(){0.0pt}1{k+2-j }{ 2} -12\genfrac(){0.0pt}1{k+1-j }{ 2} + \genfrac(){0.0pt}1{k+2-2j }{ 2} \nonumber \\&\quad + 13\genfrac(){0.0pt}1{k+1-2j }{ 2} +4\genfrac(){0.0pt}1{k-2j }{ 2} -2\genfrac(){0.0pt}1{k+1-3j }{ 2} -4\genfrac(){0.0pt}1{k-3j }{ 2} \end{aligned}$$
(A.12)
$$\begin{aligned}&N=6j+2 \nonumber \\ p(6k,3,6j+2)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}+\genfrac(){0.0pt}1{k }{ 2}- 6\genfrac(){0.0pt}1{k+1-j }{ 2}-12\genfrac(){0.0pt}1{k-j }{ 2} + 12\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 6\genfrac(){0.0pt}1{k-1-2j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2} -4\genfrac(){0.0pt}1{k-2-3j }{ 2}-\genfrac(){0.0pt}1{k-3-3j }{ 2} \end{aligned}$$
(A.13)
$$\begin{aligned} p(6k+1,3,6j+2)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 5\genfrac(){0.0pt}1{k+1 }{ 2} - 9\genfrac(){0.0pt}1{k+1-j }{ 2}-9\genfrac(){0.0pt}1{k-j }{ 2} + \genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-2j }{ 2}+4\genfrac(){0.0pt}1{k-1-2j }{ 2} -\genfrac(){0.0pt}1{k-3j }{ 2} -5\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.14)
$$\begin{aligned} p(6k+2,3,6j+2)&= 2\genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}- 12\genfrac(){0.0pt}1{k+1-j }{ 2} -6\genfrac(){0.0pt}1{k-j }{ 2} + 2\genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 14\genfrac(){0.0pt}1{k-2j }{ 2} +2\genfrac(){0.0pt}1{k-1-2j }{ 2} -2\genfrac(){0.0pt}1{k-3j }{ 2} -4\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.15)
$$\begin{aligned} p(6k+3,3,6j+2)&= 3\genfrac(){0.0pt}1{k+2 }{ 2} + 3\genfrac(){0.0pt}1{k+1 }{ 2} - \genfrac(){0.0pt}1{k+2-j }{ 2} -13\genfrac(){0.0pt}1{k+1-j }{ 2} -4\genfrac(){0.0pt}1{k-j }{ 2} + 4\genfrac(){0.0pt}1{k+1-2j }{ 2}\nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-2j }{ 2} +\genfrac(){0.0pt}1{k-1-2j }{ 2} -3\genfrac(){0.0pt}1{k-3j }{ 2} -3\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.16)
$$\begin{aligned} p(6k+4,3,6j+2)&= 4\genfrac(){0.0pt}1{k+2 }{ 2} + 2\genfrac(){0.0pt}1{k+1 }{ 2} - 2\genfrac(){0.0pt}1{k+2-j }{ 2} -14\genfrac(){0.0pt}1{k+1-j }{ 2} -2\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad + 6\genfrac(){0.0pt}1{k+1-2j }{ 2} + 12\genfrac(){0.0pt}1{k-2j }{ 2} -4\genfrac(){0.0pt}1{k-3j }{ 2} -2\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.17)
$$\begin{aligned} p(6k+5,3,6j+2)&= 5\genfrac(){0.0pt}1{k+2 }{ 2} + \genfrac(){0.0pt}1{k+1 }{ 2}- 4\genfrac(){0.0pt}1{k+2-j }{ 2}-13\genfrac(){0.0pt}1{k+1-j }{ 2}-\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad + 9\genfrac(){0.0pt}1{k+1-2j }{ 2} + 9\genfrac(){0.0pt}1{k-2j }{ 2} -5\genfrac(){0.0pt}1{k-3j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.18)
$$\begin{aligned}&N=6j+3 \nonumber \\ p(6k,3,6j+3)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}+\genfrac(){0.0pt}1{k }{ 2} - 4\genfrac(){0.0pt}1{k+1-j }{ 2}-13\genfrac(){0.0pt}1{k-j }{ 2} -\genfrac(){0.0pt}1{k-1-j }{ 2} \nonumber \\&\quad + 6\genfrac(){0.0pt}1{k-2j }{ 2} + 12\genfrac(){0.0pt}1{k-1-2j }{ 2} -3\genfrac(){0.0pt}1{k-1-3j }{ 2} -3\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.19)
$$\begin{aligned} p(6k+1,3,6j+3)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 5\genfrac(){0.0pt}1{k+1 }{ 2}- 6\genfrac(){0.0pt}1{k+1-j }{ 2} -12\genfrac(){0.0pt}1{k-j }{ 2} + 9\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 9\genfrac(){0.0pt}1{k-1-2j }{ 2} -4\genfrac(){0.0pt}1{k-1-3j }{ 2} -2\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.20)
$$\begin{aligned} p(6k+2,3,6j+3)&= 2\genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}- 9\genfrac(){0.0pt}1{k+1-j }{ 2}-9\genfrac(){0.0pt}1{k-j }{ 2} + 12\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 6\genfrac(){0.0pt}1{k-1-2j }{ 2} -5\genfrac(){0.0pt}1{k-1-3j }{ 2} -\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.21)
$$\begin{aligned} p(6k+3,3,6j+3)&= 3\genfrac(){0.0pt}1{k+2 }{ 2} + 3\genfrac(){0.0pt}1{k+1 }{ 2} - 12\genfrac(){0.0pt}1{k+1-j }{ 2}-6\genfrac(){0.0pt}1{k-j }{ 2} + \genfrac(){0.0pt}1{k+1-2j }{ 2}+ 13\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad +4\genfrac(){0.0pt}1{k-1-2j }{ 2} -\genfrac(){0.0pt}1{k-3j }{ 2} -4\genfrac(){0.0pt}1{k-1-3j }{ 2}-\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.22)
$$\begin{aligned} p(6k+4,3,6j+3)&= 4\genfrac(){0.0pt}1{k+2 }{ 2} + 2\genfrac(){0.0pt}1{k+1 }{ 2}- \genfrac(){0.0pt}1{k+2-j }{ 2}-13\genfrac(){0.0pt}1{k+1-j }{ 2} -4\genfrac(){0.0pt}1{k-j }{ 2}+ 2\genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 14\genfrac(){0.0pt}1{k-2j }{ 2} +2\genfrac(){0.0pt}1{k-1-2j }{ 2} -\genfrac(){0.0pt}1{k-3j }{ 2} -5\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.23)
$$\begin{aligned} p(6k+5,3,6j+3)&= 5\genfrac(){0.0pt}1{k+2 }{ 2} + \genfrac(){0.0pt}1{k+1 }{ 2}- 2\genfrac(){0.0pt}1{k+2-j }{ 2}-14\genfrac(){0.0pt}1{k+1-j }{ 2} -2\genfrac(){0.0pt}1{k-j }{ 2} + 4\genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-2j }{ 2} +\genfrac(){0.0pt}1{k-1-2j }{ 2} -2\genfrac(){0.0pt}1{k-3j }{ 2} -4\genfrac(){0.0pt}1{k-1-3j }{ 2} \end{aligned}$$
(A.24)
$$\begin{aligned}& N=6j+4 \nonumber \\ p(6k,3,6j+4)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}+\genfrac(){0.0pt}1{k }{ 2} - 2\genfrac(){0.0pt}1{k+1-j }{ 2} -14\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad -2\genfrac(){0.0pt}1{k-1-j }{ 2} + 2\genfrac(){0.0pt}1{k-2j }{ 2} + 14\genfrac(){0.0pt}1{k-1-2j }{ 2} +2\genfrac(){0.0pt}1{k-2-2j }{ 2} \nonumber \\&\quad -\genfrac(){0.0pt}1{k-1-3j }{ 2}-4\genfrac(){0.0pt}1{k-2-3j }{ 2}-\genfrac(){0.0pt}1{k-3-3j }{ 2} \end{aligned}$$
(A.25)
$$\begin{aligned} p(6k+1,3,6j+4)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 5\genfrac(){0.0pt}1{k+1 }{ 2} - 4\genfrac(){0.0pt}1{k+1-j }{ 2} -13\genfrac(){0.0pt}1{k-j }{ 2} -\genfrac(){0.0pt}1{k-1-j }{ 2} + 4\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-1-2j }{ 2} +\genfrac(){0.0pt}1{k-2-2j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2} -5\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.26)
$$\begin{aligned} p(6k+2,3,6j+4)&= 2\genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2}- 6\genfrac(){0.0pt}1{k+1-j }{ 2}-12\genfrac(){0.0pt}1{k-j }{ 2} + 6\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 12\genfrac(){0.0pt}1{k-1-2j }{ 2} -2\genfrac(){0.0pt}1{k-1-3j }{ 2} -4\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.27)
$$\begin{aligned} p(6k+3,3,6j+4)&= 3\genfrac(){0.0pt}1{k+2 }{ 2} + 3\genfrac(){0.0pt}1{k+1 }{ 2}- 9\genfrac(){0.0pt}1{k+1-j }{ 2} -9\genfrac(){0.0pt}1{k-j }{ 2} + 9\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 9\genfrac(){0.0pt}1{k-1-2j }{ 2} -3\genfrac(){0.0pt}1{k-1-3j }{ 2} -3\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.28)
$$\begin{aligned} p(6k+4,3,6j+4)&= 4\genfrac(){0.0pt}1{k+2 }{ 2} + 2\genfrac(){0.0pt}1{k+1 }{ 2}- 12\genfrac(){0.0pt}1{k+1-j }{ 2} -6\genfrac(){0.0pt}1{k-j }{ 2}+ 12\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 6\genfrac(){0.0pt}1{k-1-2j }{ 2} -4\genfrac(){0.0pt}1{k-1-3j }{ 2} -2\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.29)
$$\begin{aligned} p(6k+5,3,6j+4)&= 5\genfrac(){0.0pt}1{k+2 }{ 2} + \genfrac(){0.0pt}1{k+1 }{ 2}- \genfrac(){0.0pt}1{k+2-j }{ 2} -13\genfrac(){0.0pt}1{k+1-j }{ 2} -4\genfrac(){0.0pt}1{k-j }{ 2}+ \genfrac(){0.0pt}1{k+1-2j }{ 2} \nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-2j }{ 2} +4\genfrac(){0.0pt}1{k-1-2j }{ 2} -5\genfrac(){0.0pt}1{k-1-3j }{ 2} -\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.30)
$$\begin{aligned}& N=6j+5 \nonumber \\ p(6k,3,6j+5)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2} + \genfrac(){0.0pt}1{k }{ 2}- \genfrac(){0.0pt}1{k+1-j }{ 2}-13\genfrac(){0.0pt}1{k-j }{ 2}-4\genfrac(){0.0pt}1{k-1-j }{ 2}\nonumber \\&\quad +12\genfrac(){0.0pt}1{k-1-2j }{ 2} + 6\genfrac(){0.0pt}1{k-2-2j }{ 2} -3\genfrac(){0.0pt}1{k-2-3j}{ 2} -3\genfrac(){0.0pt}1{k-3-3j }{ 2} \end{aligned}$$
(A.31)
$$\begin{aligned} p(6k+1,3,6j+5)&= \genfrac(){0.0pt}1{k+2 }{ 2} + 5\genfrac(){0.0pt}1{k+1 }{ 2} -2\genfrac(){0.0pt}1{k+1-j }{ 2} -14\genfrac(){0.0pt}1{k-j }{ 2} \nonumber \\&\quad -2\genfrac(){0.0pt}1{k-1-j }{ 2}+\genfrac(){0.0pt}1{k-2j }{ 2} +13\genfrac(){0.0pt}1{k-1-2j }{ 2} \nonumber \\&\quad +4\genfrac(){0.0pt}1{k-2-2j }{ 2}-4\genfrac(){0.0pt}1{k-2-3j }{ 2} -2\genfrac(){0.0pt}1{k-3-3j }{ 2} \end{aligned}$$
(A.32)
$$\begin{aligned} p(6k+2,3,6j+5)&= 2\genfrac(){0.0pt}1{k+2 }{ 2} + 4\genfrac(){0.0pt}1{k+1 }{ 2} - 4\genfrac(){0.0pt}1{k+1-j }{ 2}-13\genfrac(){0.0pt}1{k-j }{ 2} -\genfrac(){0.0pt}1{k-1-j }{ 2} \nonumber \\&\quad + 2\genfrac(){0.0pt}1{k-2j }{ 2} + 14\genfrac(){0.0pt}1{k-1-2j }{ 2} +2\genfrac(){0.0pt}1{k-2-2j }{ 2} \nonumber \\&\quad -5\genfrac(){0.0pt}1{k-2-3j }{ 2} -\genfrac(){0.0pt}1{k-3-3j }{ 2} \end{aligned}$$
(A.33)
$$\begin{aligned} p(6k+3,3,6j+5)&= 3\genfrac(){0.0pt}1{k+2 }{ 2} + 3\genfrac(){0.0pt}1{k+1 }{ 2} - 6\genfrac(){0.0pt}1{k+1-j }{ 2}-12\genfrac(){0.0pt}1{k-j }{ 2}+ 4\genfrac(){0.0pt}1{k-2j }{ 2}\nonumber \\&\quad + 13\genfrac(){0.0pt}1{k-1-2j }{ 2} +\genfrac(){0.0pt}1{k-2-2j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2}\nonumber \\&\quad -4\genfrac(){0.0pt}1{k-2-3j }{ 2}-\genfrac(){0.0pt}1{k-3-3j }{ 2} \end{aligned}$$
(A.34)
$$\begin{aligned} p(6k+4,3,6j+5)&= 4\genfrac(){0.0pt}1{k+2 }{ 2} + 2\genfrac(){0.0pt}1{k+1 }{ 2}- 9\genfrac(){0.0pt}1{k+1-j }{ 2}-9\genfrac(){0.0pt}1{k-j }{ 2} + 6\genfrac(){0.0pt}1{k-2j }{ 2} \nonumber \\&\quad + 12\genfrac(){0.0pt}1{k-1-2j }{ 2} -\genfrac(){0.0pt}1{k-1-3j }{ 2} -5\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.35)
$$\begin{aligned} p(6k+5,3,6j+5)&= 5\genfrac(){0.0pt}1{k+2 }{ 2} + \genfrac(){0.0pt}1{k+1 }{ 2} - 12\genfrac(){0.0pt}1{k+1-j }{ 2} -6\genfrac(){0.0pt}1{k-j }{ 2}+ 9\genfrac(){0.0pt}1{k-2j }{ 2}\nonumber \\&\quad + 9\genfrac(){0.0pt}1{k-1-2j }{ 2} -2\genfrac(){0.0pt}1{k-1-3j }{ 2} -4\genfrac(){0.0pt}1{k-2-3j }{ 2} -4\genfrac(){0.0pt}1{k-2-3j }{ 2} \end{aligned}$$
(A.36)

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Castillo, A., Flores, S., Hernandez, A. et al. Quasipolynomials and Maximal Coefficients of Gaussian Polynomials. Ann. Comb. 23, 589–611 (2019). https://doi.org/10.1007/s00026-019-00467-2

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