8
$\begingroup$

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 distinct parts and let $d_t(n)$ be the number of such partitions that are $t$-cores. Then it is easy to see $d_2(n)=c_2(n)$.

QUESTION. Is this true? $$\sum_{n\geq0}d_3(n)\,q^n=\sum_{k\geq0}q^{k^2} +\sum_{k\geq1}q^{2\binom{k+1}2}.$$

Note that I have simplified the generating function from $$\frac12\prod_{n\geq1}(1-q^{2n})(1+q^{2n-1})^2+\prod_{n\geq1}(1-q^{2n})(1+q^{2n})^2-\frac12.$$

$\endgroup$

1 Answer 1

13
$\begingroup$

The set of $3$-core partitions can be described explicitly.

Theorem The partition $\lambda=\{\lambda_1,\lambda_2,\dots\}$ of length $k$ (that is, $\lambda_k > 0$ but $\lambda_{k+1} = \lambda_{k+2} = \cdots = 0$) is a $3$-core if and only if the sequence of differences $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots,\lambda_k - \lambda_{k+1}\}$ is of the form $\{2,2,\dots,2,1,0,1,0,\dots ,1\}$ or $\{2,2,\dots,2,0,1,0,1,\dots ,1\}$.

Proof: It is easy to check by hand that a $3$-hook appears in the situations where

  • a) some member of the sequence is $\geq 3$,

  • b) two members of the sequence in a row are $0$'s,

  • c) there is a $1$ in the sequence that is not followed by a $0$,

  • d) there is a $0$ in the sequence that is not followed by a $1$.

These correspond to the four possible shapes of a $3$ hook-strip in the boundary. If all of these patterns are avoided, then the partition has no hooks of length $3$.

Therefore the partitions with distinct parts that are $3$-cores have a difference sequence $\{2,2,\dots,2\}$ or $\{2,2,\dots,2,1\}$. The size of the partitions in the first case are given by $2\binom{k+1}{2}$ and the sizes in the second case are given by $k^2$, where $k\geq 1$, and this implies your generating function.

$\endgroup$
2
  • 1
    $\begingroup$ +1, nice work! I had no idea that any $r$-cores with $r \geq 2$ could be characterized in such an explicit manner. $\endgroup$ Oct 14, 2018 at 17:09
  • 2
    $\begingroup$ This matches the characterization of 3-core partitions by Neville Robbins in "On t-core partitions" Fibonacci Quarterly 38 (2000) 39--48. He does not give a similar result for 4- (or higher) core partitions, suggesting they may not be as nicely described. $\endgroup$ Oct 14, 2018 at 20:35

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.