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

Theorem The partition $\lambda=\{\lambda_1,\lambda_2,\dots\}$ is a $3$ core if and only if the sequence of differences $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots\}$ 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 $0$ 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.

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.

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

Theorem The partition $\lambda=\{\lambda_1,\lambda_2,\dots\}$ is a $3$ core if and only if the sequence of differences $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots\}$ 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) three members of the sequence in a row are $0$'s c) there is a $1$ in the sequence that is not followed by $0$ 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.

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

Theorem The partition $\lambda=\{\lambda_1,\lambda_2,\dots\}$ is a $3$ core if and only if the sequence of differences $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots\}$ 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 $0$ 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.

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

Theorem The partition $\lambda=\{\lambda_1,\lambda_2,\dots\}$ is a $3$ core if and only if the sequence of differences $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots\}$ is of the form $\{2,2,\dots,2,1,0,1,0,\dots\}$ or $\{2,2,\dots,2,0,1,0,1,\dots\}$.

Proof: It is easy to check by hand that a $3$-hook appears in the situations where a) some member of the sequence is $3$ b) three members of the sequence in a row are $0$'s c) there is a $1$ in the sequence that is not followed by $0$ 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.

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

Theorem The partition $\lambda=\{\lambda_1,\lambda_2,\dots\}$ is a $3$ core if and only if the sequence of differences $\{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots\}$ 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) three members of the sequence in a row are $0$'s c) there is a $1$ in the sequence that is not followed by $0$ 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.