Schur’s partition theorem, companions, refinements and generalizations
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- by Krishnaswami Alladi and Basil Gordon PDF
- Trans. Amer. Math. Soc. 347 (1995), 1591-1608 Request permission
Abstract:
Schur’s partition theorem asserts the equality $S(n) = {S_1}(n)$, where $S(n)$ is the number of partitions of $n$ into distinct parts $\equiv 1,2(\mod 3)$ and ${S_1}(n)$ is the number of partitions of $n$ into parts with minimal difference $3$ and no consecutive multiples of $3$. Using a computer search Andrews found a companion result $S(n) = {S_2}(n)$, where ${S_2}(n)$ is the number of partitions of $n$ whose parts ${e_i}$ satisfy ${e_i} - {e_{i + 1}} \geqslant 3,2or5$ according as ${e_i} \equiv 1,2$ or $(\bmod 3)$. By means of a new technique called the method of weighted words, a combinatorial as well as a generating function proof of both these theorems are given simultaneously. It is shown that ${S_1}(n)$ and ${S_2}(n)$ are only two of six companion partition functions ${S_j}(n),j = 1,2, \ldots 6$, all equal to $S(n)$. A three parameter refinement and generalization of these results is obtained.References
- 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
- 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 —, Refinements and generalizations of Capparelli’s conjecture on partitions, J. Algebra (to appear).
- 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 —, On Schur’s second partition theorem, Glasgow Math. J. 9 (1967), 127-132. —, A new generalization of Schur’s second partition theorem, Acta Arith. 4 (1968), 429-434. —, A general partition theorem with difference conditions, Amer. J. Math. 191 (1969), 18-24.
- George E. Andrews, The use of computers in search of identities of the Rogers-Ramanujan type, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 377–387. MR 0316373 —, Schur’s theorem, Capparelli’s conjecture and $q$-trinomial coefficients, The Rademacher Legacy to Mathematics (Proc. Rademacher Centenary Conf, 1992), Contemp. Math., Amer. Math. Soc., Providence, RI, 1994, pp. 141-154.
- D. M. Bressoud, On a partition theorem of Göllnitz, J. Reine Angew. Math. 305 (1979), 215–217. MR 518863, DOI 10.1515/crll.1979.305.215
- 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
- Stefano Capparelli, On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra 154 (1993), no. 2, 335–355. MR 1206124, DOI 10.1006/jabr.1993.1017
- H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154–190 (German). MR 211973, DOI 10.1515/crll.1967.225.154 I. J. Schur, Zur additiven Zahlentheorie, Gessammelte Abhandlungen, vol. 2, Springer-Verlag, Berlin, 1973, pp. 43-50.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 1591-1608
- MSC: Primary 11P83; Secondary 05A17, 05A19, 11P81
- DOI: https://doi.org/10.1090/S0002-9947-1995-1297520-X
- MathSciNet review: 1297520