Geodesic
On Stanley's partition function
The electronic journal of combinatorics, Tome 17 (2010)
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Stanley defined a partition function $t(n)$ as the number of partitions $\lambda$ of $n$ such that the number of odd parts of $\lambda$ is congruent to the number of odd parts of the conjugate partition $\lambda'$ 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.
DOI : 10.37236/480
Classification : 05A10, 05A15, 05A17
Mots-clés : partition function, generating function 
@article{10_37236_480,
     author = {William Y. C. Chen and Kathy Q. Ji and Albert J. W. Zhu},
     title = {On {Stanley's} partition function},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/480},
     zbl = {1205.05007},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/480/}
}
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%A Kathy Q. Ji
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William Y. C. Chen; Kathy Q. Ji; Albert J. W. Zhu. On Stanley's partition function. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/480

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