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On Stanley's Partition Function

By William Y. C. Chen, Kathy Q. Ji and Albert J. W. Zhu


Stanley defined a partition function t(n) as the number of partitions λ of n such that the number of odd parts of λ is congruent to the number of odd parts of the conjugate partition λ ′ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n) − t(n). As a consequence, we see that t(n) has the same parity as the ordinary partition function p(n). A simple combinatorial explanation of this fact is also provided

Year: 2010
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