A combinatorial correspondence related to Göllnitz’ (big) partition theorem and applications

Alladi, Krishnaswami

doi:10.1090/S0002-9947-97-01944-2

A combinatorial correspondence related to Göllnitz’ (big) partition theorem and applications
HTML articles powered by AMS MathViewer

by Krishnaswami Alladi PDF

Trans. Amer. Math. Soc. 349 (1997), 2721-2735 Request permission

Abstract:

In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi’s triple product identity and Schur’s partition theorem.

References

K. Alladi, Partition identities involving gaps and weights, Trans. Amer. Math. Soc. (to appear).
K. Alladi, and G. E. Andrews, A new key identity for Göllnitz’ (big) partition theorem in Proc. 10$^{\text {th}}$ Anniv. Conf. Ramanujan Math. Soc., Contemp. Math. (to appear).
Krishnaswami Alladi, George E. Andrews, and Basil Gordon, Generalizations and refinements of a partition theorem of Göllnitz, J. Reine Angew. Math. 460 (1995), 165–188. MR 1316576
Krishnaswami Alladi and Basil Gordon, Generalizations of Schur’s partition theorem, Manuscripta Math. 79 (1993), no. 2, 113–126. MR 1216769, DOI 10.1007/BF02568332
George E. Andrews, On a partition theorem of Göllnitz and related formulae, J. Reine Angew. Math. 236 (1969), 37–42. MR 248101, DOI 10.1515/crll.1969.236.37
George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
George E. Andrews, $q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 858826, DOI 10.1090/cbms/066
David M. Bressoud, A combinatorial proof of Schur’s 1926 partition theorem, Proc. Amer. Math. Soc. 79 (1980), no. 2, 338–340. MR 565367, DOI 10.1090/S0002-9939-1980-0565367-X
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154–190 (German). MR 211973, DOI 10.1515/crll.1967.225.154
Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891, DOI 10.1007/978-3-642-61947-2
J. J. Sylvester, A constructive theory of partitions arranged in three Acts, an Interact and an Exodion, Amer. J. Math. 5 (1882), 251-330.