A unification of two refinements of Euler’s partition theorem. (English) Zbl 1218.05020
Summary: We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.
Keywords:
partition; Euler’s partition theorem; refinement; bijection; Bessenrodt’s bijection; Andrews-Olsson’s theoremReferences:
| [1] | Alder, H.L.: Partition identities-from Euler to the present. Am. Math. Mon. 76, 733–746 (1969) · Zbl 0186.01603 · doi:10.2307/2317861 |
| [2] | Andrews, G.E.: On generalizations of Euler’s partition theorem. Mich. Math. J. 1, 491–498 (1966) · Zbl 0149.01902 |
| [3] | Andrews, G.E.: The Theory of Partitions. Addison–Wesley, Reading (1976) · Zbl 0371.10001 |
| [4] | Andrews, G.E.: On a partition theorem of N.J. Fine. J. Natl. Acad. Math. India 1, 105–107 (1983) · Zbl 0579.05008 |
| [5] | Andrews, G.E.: On a partition function of Richard Stanley. Electron. J. Comb. 11(2), #R1 (2004) · Zbl 1067.05005 |
| [6] | Andrews, G.E., Eriksson, K.: Integer Partitions. Cambridge Univ. Press, Cambridge (2004) · Zbl 1073.11063 |
| [7] | Andrews, G.E., Olsson, J.B.: Partition identities with an application to group representation theory. J. Reine Angew. Math. 413, 198–212 (1991) · Zbl 0704.20003 |
| [8] | Bessenrodt, C.: A combinatorial proof of a refinement of the Andrews-Olsson partition identity. Eur. J. Comb. 12, 271–276 (1991) · Zbl 0743.05003 |
| [9] | Bessenrodt, C.: A bijection for Lebesgue’s partition identity in the spirit of Sylvester. Discrete Math. 132, 1–10 (1994) · Zbl 0811.05003 · doi:10.1016/0012-365X(94)90228-3 |
| [10] | Bessenrodt, C.: Generalizations of the Andrews-Olsson partition identity. Discrete Math. 141, 11–22 (1995) · Zbl 0839.05009 · doi:10.1016/0012-365X(93)E0189-B |
| [11] | Boulet, C.: A four parameter partition identity. Ramanujan J. 12(3), 315–320 (2006) · Zbl 1114.05008 · doi:10.1007/s11139-006-0145-4 |
| [12] | Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions. Ramanujan J. 1, 101–111 (1997) · Zbl 0909.05008 · doi:10.1023/A:1009771306380 |
| [13] | Bousquet-Mélou, M., Eriksson, K.: Lecture hall partitions II. Ramanujan J. 1, 165–185 (1997) · Zbl 0909.05009 · doi:10.1023/A:1009768118404 |
| [14] | Bressoud, D.: Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge Univ. Press, Cambridge (1999) · Zbl 0944.05001 |
| [15] | Fine, N.J.: Basic Hypergeometric Series and Applications. Math. Surveys, vol. 27. AMS, Providence (1988) · Zbl 0647.05004 |
| [16] | Glaisher, J.W.L.: A theorem in partitions. Messenger Math. 12, 158–170 (1883) |
| [17] | Kim, D., Yee, A.J.: A note on partitions into distinct parts and odd parts. Ramanujan J. 3, 227–231 (1999) · Zbl 0930.05006 · doi:10.1023/A:1006905826378 |
| [18] | Pak, I.: On Fine’s partition theorems, Dyson, Andrews, and missed opportunities. Math. Intell. 25, 10–16 (2003) · doi:10.1007/BF02985633 |
| [19] | Sylvester, J.: A constructive theory of partitions, arranged in three acts, an interact and an exodion. Am. J. Math. 5, 251–330 (1882) · JFM 15.0129.02 · doi:10.2307/2369545 |
| [20] | Yee, A.J.: A combinatorial proof of Andrews’ partition functions related to Schur’s partition theorem. Proc. Am. Math. Soc. 130, 2229–2235 (2002) · Zbl 0989.05008 · doi:10.1090/S0002-9939-02-06560-7 |
| [21] | Zeng, J.: The q-variations of Sylvester’s bijection between odd and strict partitions. Ramanujan J. 9(3), 289–303 (2005) · Zbl 1085.05014 · doi:10.1007/s11139-005-1869-2 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
