On the $m$-ary partition numbers

Yasuyuki Kachi ; Pavlos Tzermias

Algebra and discrete mathematics, Tome 19 (2015) no. 1, pp. 67-76.

Résumé

We give explicit formulas and recurrences for the $m$-ary partition numbers and relate them to Toeplitz-Hessenberg determinants. Some of these results are direct analogues of similar statements for the classical (unrestricted) partition numbers.

Détail

Keywords: $m$-ary partition numbers, generating functions, Toeplitz-Hessenberg determinants.

Bibliographie

[1] G. Alkauskas, “Congruence properties of the function that counts compositions into powers of $2$”, J. Integer Seq., 13:5 (2010), Article 10.5.3, 9 pp. | MR | Zbl

[2] G. Alkauskas, “A generalization of the Rödseth-Gupta theorem on binary partitions”, Lithuanian Math. J., 43:2 (2003), 103–110 | DOI | MR | Zbl

[3] J.-P. Allouche, J. Shallit, “The ubiquitous Prouhet-Thue-Morse sequence”, Sequences and their aplications, Proceedings of SETA 98, eds. Ding, C. Helleseth, T., Niederreiter, H., Springer-Verlag, 1999, 1–16 | DOI | MR | Zbl

[4] G. Andrews, “Congruence properties of the $m$-ary partition function”, J. Number Theory, 3 (1971), 104–110 | DOI | MR | Zbl

[5] B. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, New York, 1994 | MR | Zbl

[6] F. Brioschi, “Sulle funzioni Bernoulliane ed Euleriane”, Annali di Matematica Pura ed Applicada, 1858, 260–263 | DOI

[7] J.-H. Bruinier, K. Ono, “Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms”, Adv. Math., 246 (2013), 198–219 | DOI | MR | Zbl

[8] N. Calkin, H. Wilf, “Recounting the rationals”, Amer. Math. Monthly, 107:4 (2007), 360–363 | DOI | MR

[9] K.-W. Chen, “Inversion of generating functions using determinants”, J. Integer Seq., 10:10 (2007), Article 07.10.5, 10 pp. | MR

[10] R.-F. Churchhouse, “Congruence properties of the binary partition function”, Proc. Cambridge Philos. Soc., 66 (1969), 371–376 | DOI | MR | Zbl

[11] L. Comtet, Advanced Combinatorics, The art of finite and infinite expansions, D. Reidel Publishing Co., Dordrecht, 1974 | MR | Zbl

[12] N.-G. de Bruijn, “On Mahler's partition problem”, Indagationes Math., 10 (1948), 210–220

[13] G. Fera, V. Talamini, Explicit formulas using partitions of integers for numbers, 2013, preprint, arXiv: http://arxiv.org/pdf/1211.1440v2.pdf

[14] E. Fergola, “Sopra lo sviluppo della funzione $1/(ce^x-1)$, e sopra una nuova espressione dei numeri di Bernoulli”, Memorie della Reale Accademia delle Scienze di Napoli, 2 (1857), 315–324

[15] P. Flajolet, R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009 | MR | Zbl

[16] D. Fuchs, S. Tabachnikov, Mathematical Omnibus, Thirty lectures on classic mathematics, American Mathematical Society, Providence, RI, 2007 | MR | Zbl

[17] H. Gupta, “A direct proof of the Churchhouse conjecture concerning binary partitions”, Indian J. Math., 18:1 (1976), 1–5 | MR | Zbl

[18] H. Gupta, “A simple proof of the Churchhouse conjecture concerning binary partitions”, Indian J. Pure Appl. Math., 3:5 (1972), 791–794 | MR | Zbl

[19] H. Gupta, “Proof of the Churchhouse conjecture concerning binary partitions”, Proc. Cambridge Philos. Soc., 70 (1971), 53–56 | DOI | MR | Zbl

[20] K. Hessenberg, Behandlung linearer Eigenwertaufgaben mit Hilfe der Hamilton-Cayleyschen Gleichung, Bericht der Reihe Numerische Verfahren, IPM, Darmstadt, 1940, 36 pp.

[21] M.-D. Hirschhorn, J.-H. Loxton, “Congruence properties of the binary partition function”, Math. Proc. Cambridge Philos. Soc., 78 (1975), 437–442 | DOI | MR | Zbl

[22] M.-D. Hirschhorn, J.-A. Sellers, “A different view of $m$-ary partitions”, Australas. J. Combin., 30 (2004), 193–196 | MR | Zbl

[23] K. Mahler, “On a special functional equation”, J. London Math. Soc., 15 (1940), 115–123 | DOI | MR

[24] K. Mahler, “Berichtigung zu der Arbeit von K. Mahler: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen”, Math. Ann., 103 (1930), 532 | DOI | MR | Zbl

[25] K. Mahler, “Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen”, Math. Ann., 101 (1929), 342–366 | DOI | MR | Zbl

[26] J. Malenfant, Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers, 2011, preprint, arXiv: http://arxiv.org/pdf/1103.1585.pdf

[27] G. Meinardus, “Asymptotische Aussagenüber Partitionen”, Math. Z., 59 (1954), 388–398 | DOI | MR | Zbl

[28] T. Muir, The Theory of Determinants in the Historical Order of Development, v. II, The period 1841 to 1860, Macmillan and Co., London, 1911 | Zbl

[29] M. Ram Murty, V. Kumar Murty, The Mathematical Legacy of Srinivasa Ramanujan, Springer, New Delhi, 2013 | Zbl

[30] W.-B. Pennington, “On Mahler's partition problem”, Annals of Math., 57 (1953), 531–546 | DOI | MR | Zbl

[31] H. Rademacher, “On the expansion of the partition function in a series”, Annals of Math., 44 (1943), 416–422 | DOI | MR | Zbl

[32] H. Rademacher, “On the partition function $p(n)$”, Proc. London Math. Soc. (2), 43 (1937), 241–254 | MR

[33] B. Richmond, “Mahler's partition problem”, Ars Combinatoria, 2 (1976), 169–189 | MR | Zbl

[34] Ö. Rödseth, “Some arithmetical properties of $m$-ary partitions”, Proc. Cambridge Philos. Soc., 68 (1970), 447–453 | DOI | MR | Zbl

[35] Ö. Rödseth, J.-A. Sellers, “Binary partitions revisited”, J. Combin. Theory Ser. A, 98:1 (2002), 33–45 | DOI | MR | Zbl

[36] Ö. Rödseth, J.-A. Sellers, “On $m$-ary partition function congruences: A fresh look at a past problem”, J. Number Theory, 87:2 (2001), 270–281 | DOI | MR | Zbl

[37] O. Toeplitz, “Zur Transformation der Scharen bilinearer Formen von unendlichvielen Veränderlichen”, Nachr. der Kgl. Gesselschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1907, 110–115 | Zbl

[38] D. Vella, “Explicit formulas for Bernoulli and Euler numbers”, Integers, 8 (2008), A01, 7 pp. | MR | Zbl

[39] H. Wilf, Lectures on Integer Partitions, Pacific Institute for the Mathematical Sciences Distinguished Chair Lectures, University of Victoria, 2000 http://www.math.upenn.edu/<nobr>$\sim$</nobr>wilf/PIMS/PIMSLectures.pdf