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Hook Length Formulas for Partitions and Plane Trees

An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths

By Guo-Niu Han

[download the paper: .ps - .pdf, 35 pages, 2008/04/04]

Abstract. We discover an explicit expansion formula for the powers s of the Euler Product (or Dedekind η-function) in terms of hook lengths of partitions, where the exponent s is any complex number. Several classical formulas have been derived for certain integers s by Euler, Jacobi, Klein, Fricke, Atkin, Winquist, Dyson and Macdonald. In particular, Macdonald obtained expansion formulas for the integer exponents s for which there exists a semi-simple Lie algebra of dimension s. For the type Al(a) he has expressed the (t2-1)-st power of the Euler Product as a sum of weighted integer vectors of length t for any integer t. Kostant has considered the general case for any positive integer s and obtained further properties.

The present paper proposes a new approach. We convert the weighted vectors of length t used by Macdonald in his identity for type Al(a) to weighted partitions with free parameter t, so that a new identity on the latter combinatorial structures can be derived without any restrictions on t. The surprise is that the weighted partitions have a very simple form in terms of hook lengths of partitions. As applications of our formula, we find some new identities about hook lengths, including the "marked hook formula". We also improve a result due to Kostant. The proof of the Main Theorem is based on Macdonald's identity for Al(a) and on the properties of a bijection between t-cores and integer vectors constructed by Garvan, Kim and Stanton.

The Main Theorem is the following hook length formula for partitions:

Summary

  1. Introduction. The Main Theorem. Selected results.
  2. Basic consequences and specializations.
    1. Equivalent forms.
    2. Corollaries.
    3. Specialization for β=0. Generating function for partitions.
    4. Specialization for β=1.
    5. Specialization for β=∞. Classical hook length formula and the Robinson-Schensted-Knuth correspondence.
    6. Specialization for β=-1.
    7. Specialization for β=2. Euler's pentagonal theorem. Example for illustrating the Main Theorem.
    8. Specialization for β=25. Ramanujan τ-function. Example for illustrating the Main Theorem.
  3. Specialization for β=4. Jacobi's triple product formula.
  4. Specialization for β=9.
  5. Proof of the Main Theorem.
    1. Fundamental properties of t-cores and V-codings.
    2. The bijection φV and an example.
    3. Proof of the first property.
    4. Proof of the second property.
    5. End of the proof of the Main Theorem.
  6. New formulas about hook lengths.
    1. Comparing the coefficients of β.
    2. Stanley-Elder-Bessenrodt-Bacher-Manivel Theorem.
    3. Comparing the coefficients of β2.
    4. Comparing the coefficients of &betanxn and &betan-1xn. The marked hook formula.
    5. Comparing the coefficients of &betan-2xn and of &betan-3xn.
  7. Improvement of a result due to Kostant.
  8. The magic partition formula.
  9. Reversion of the Euler Product.
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