3
$\begingroup$

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$.

We call $\lambda$ a $t$-core partition if none of its hooks $h_u$ equals $t$. Define $c_t(n)$ to be the number of partitions of $n$ that are $t$-core partitions. It's well-known that $$\sum_{n\geq0}c_t(n)\,q^n=\prod_{k=1}^{\infty}\frac{(1-q^{tk})^t}{1-q^k}.$$ For example, $\sum_{n=0}^{\infty}c_2(n)\,q^n=\sum_{k=0}^{\infty}q^{\binom{k+1}2}$.

Now, consider only those partitions of $n$ with odd parts and let $O_t(n)$ be the number of such partitions that are $t$-cores.

QUESTION. Is this true? $$\sum_{n\geq1}O_3(n)\,q^n=(1+q)\sum_{k\geq1}q^{k^2}.$$

$\endgroup$

1 Answer 1

9
$\begingroup$

Suppose the partition with $k$ parts $\lambda=\{\lambda_1\geq \lambda_2\geq\dots\geq\lambda_k\geq \lambda_{k+1}=0\}$ is a partition with only odd parts. Then we have $\lambda_i-\lambda_{i+1}$ is even for all $i\leq k-1$ and odd for $i=k$. Using the same characterization I mentioned in the previous answer we see that the sequence $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots,\lambda_{k}-\lambda_{k+1}\}$ for such a $3$-core has to be equal to $\{2,2,\dots,2,1\}$ or $\{2,2,\dots,2,0,1\}$. The first case is a partition with size $k^2$ and the second has size $1+k^2$ which implies the generating function $$\sum_{n\geq1}O_3(n)\,q^n=\sum_{k\geq1}(q^{k^2}+q^{k^2+1}).$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.