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Register a new userDistribution properties for t-hooks in partitions
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Partitions, the partition function $p(n)$, and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory and representation theory. For positive integers $n$ and $t$, we study $p_t^e(n)$ (resp. $p_t^o(n)$), the number of partitions of $n$ with an even (resp. odd) number of $t$-hooks. We study the limiting behavior of the ratio $p_t^e(n)/p(n)$, which also gives $p_t^o(n)/p(n)$ since $p_t^e(n) + p_t^0(n) = p(n)$. For even $t$, we show that $$limlimits_{n to infty} dfrac{p_t^e(n)}{p(n)} = dfrac{1}{2},$$ and for odd $t$ we establish the non-uniform distribution $$limlimits_{n to infty} dfrac{p^e_t(n)}{p(n)} = begin{cases} dfrac{1}{2} + dfrac{1}{2^{(t+1)/2}} & text{if } 2 mid n, dfrac{1}{2} - dfrac{1}{2^{(t+1)/2}} & text{otherwise.} end{cases}$$ Using the Rademacher circle method, we find an exact formula for $p_t^e(n)$ and $p_t^o(n)$, and this exact formula yields these distribution properties for large $n$. We also show that for sufficiently large $n$, the signs of $p_t^e(n) - p_t^o(n)$ are periodic.
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