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Method for constructing bijections for classical partition identities.

A M Garsia1, S C Milne

  • 1Department of Mathematics, University of California at San Diego, La Jolla, California 92093.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 1981
PubMed
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We established a method to link two types of integer partitions: those with parts congruent to 1 or 4 (mod 5) and those with parts differing by at least 2. This advances understanding of partition theory and related identities.

Area of Science:

  • Number Theory
  • Combinatorics
  • Partition Theory

Background:

  • Integer partitions are fundamental in combinatorics.
  • The Rogers-Ramanujan identities relate specific types of partitions.
  • Understanding bijections reveals deeper structural connections.

Purpose of the Study:

  • To construct a novel bijection between two distinct partition sets.
  • To demonstrate a generalizable method for proving partition identities.
  • To connect existing bijections using a new principle.

Main Methods:

  • A cut-and-paste procedure was employed to transform partitions.
  • The construction combines Schur's bijection with two new bijections.
  • A principle of involutions was utilized to link the bijections.

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Main Results:

  • A bijection is established between partitions of n with parts congruent to 1 or 4 (mod 5) and partitions of n with parts differing by at least 2.
  • The method provides a constructive proof for this partition identity.
  • The approach is adaptable for other Rogers-Ramanujan type identities.

Conclusions:

  • The developed bijection offers new insights into partition theory.
  • The technique facilitates the construction of bijections for related identities, including Gordon's identities.
  • This work provides a unified framework for exploring partition identities.