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A Bijective Toolkit for Signed Partitions
William J. Keith
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA
Annals of Combinatorics 15 (1) pp.95-117 January, 2011
AMS Subject Classification: 05A17; 11P83
The recently formalized idea of signed partitions is examined with intent to expand the standard repertoire of mappings and statistics used in bijective proofs for ordinary partition identities. A new family of partitions is added to Schur's Theorem and observations are made concerning the behavior of signed partitions of zero in arithmetic progression.
Keywords: partitions, signed partitions, partitions of zero, complement, Ferrers diagram


1. Alladi, K., Andrews, G.E., Gordon, B.: Generalizations and refinements of a partition theorem of G¨ollnitz. J. Reine Angew. Math. 460, 165--–188 (1995)

2. Andrews, G.E.: Euler's “De Partitio Numerorum”. Bull. Amer. Math. Soc. 44(4), 561--–573 (2007)

3. Andrews, G.E.: Schur's Theorem, Partitions with Odd Parts and the Al-Salam-Carlitz Polynomials. Contemp. Math. 254, Amer. Math. Soc., Providence, RI (2000)

4. Andrews, G.E.: Two theorems of Gauss and allied identities proved arithmetically. Pacific J. Math. 41, 563--–578 (1972)

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

6. Bessenrodt, C.: A bijection for Lebesgue's partition identity in the spirit of Sylvester. Discrete Math. 132(1-3), 1--–10 (1994)

7. Garrett, K.C.: A determinant identity that implies Rogers-Ramanujan. Electron. J. Combin. 12, #35 (2005)

8. Keith, W.J.: Ranks of Partitions and Durfee Symbols. Ph.D. Thesis, Pennsylvania State University, State College (2007)

9. Pak, I.: Partition bijections, a survey. Ramanujan J. 12(1), 5--–75 (2006)

10. Stockhofe, D.: Bijektive Abbildungen auf der Menge der partitionen einer natürlichen Zahl. Bayreuth. Math. Schr. 10, 1--–59 (1982)

11. Yee, A.J.: A combinatorial proof of Andrews' partition functions related to Schur's partition theorem. Proc. Amer. Math. Soc. 130(8), 2229--–2235 (2002)

12. Zelobenko, D.P.: Compact Lie Groups and Their Representations. Translations of Mathematical Monographs Vol. 40. Amer. Math. Soc., Providence, R.I. (1973)

13. Zeng, J.: The q-variations of Sylvester's bijection between odd and strict partitions. Ramanujan J. 9(3), 289--–303 (2005)