The Reciprocity Method

  • Anthony Mendes
  • Jeffrey Remmel
Part of the Developments in Mathematics book series (DEVM, volume 43)


In previous chapters, we defined ring homomorphisms \(\varphi\) on the ring of symmetric functions in order to find generating functions for permutation statistics. Doing this requires an a priori insightful definition of what the function \(\varphi\) should be. This chapter introduces the reciprocity method, which provides a way to come up with such a function \(\varphi\).


Reciprocity Method Permutation Statistics Descent Pair Pattern-avoiding Permutations Generating Function Manipulations 
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  1. 64.
    Jones, M.E., Remmel, J.: Generating functions for the number of permutations with no consecutive occurrences of 1p23(p − 1) or 13(p − 1)2p. Pure Math. Appl. (to appear)Google Scholar
  2. 65.
    Jones, M.E., Remmel, J.: Pattern matching in the cycle structures of permutations. Pure Math. Appl. 22(2), 173–208 (2011)MathSciNetzbMATHGoogle Scholar
  3. 66.
    Jones, M.E., Remmel, J.: A reciprocity approach to computing generating functions for permutations with no pattern matches. In: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011). Discrete Mathematics and Theoretical Computer Science Proceedings, AO, pp. 551–562. Association of Discrete Mathematics and Theoretical Computer Science, Nancy (2011)Google Scholar
  4. 67.
    Jones, M.E., Remmel, J.B.: Applying a reciprocity method to count permutations avoiding two families of consecutive patterns. Pure Math. Appl. 24(2), 151–178 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Anthony Mendes
    • 1
  • Jeffrey Remmel
    • 2
  1. 1.Mathematics DepartmentCalifornia Polytechnic State UniversitySan Luis ObispoUSA
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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