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edit approved Mar 26, 2014 at 6:39

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Here's a more self-contained description of this module. For simplicity, I'll consider only the case $n=2m$. Consider the vector space $V$ spanned by the set $P$ of ordered partitions of $[2m]$ into $m$ blocks of size two. The symmetric group $S_{2m}$ acts naturally on $V$. Now let $T$ be the set of ordered partitions of $[2m]$ with $m-2$ blocks of size two and one block of size 4, and for $t\in T$, let $\alpha(t)$ be the sum in $V$ of all the elements of $P$ from which $t$ can be obtained by merging two adjacent blocks. TheThen the quotient of $V$ by the $S_n$-module spanned by the $\alpha(t)$ is the desired $S_{2m}$-module, with dimension $E_{2m}$$A_{2m}$.

For example, with $n=4$, there are 6 ordered partitions:

12 | 34, 13 | 24, 14 | 23, 34 | 12, 24 | 13, and 23 | 14.

There is only one element $t\in T$, and $\alpha(t)$ is the sum of all six ordered partitions in $P$, so the quotient module has dimension $5=A_4$.

Instead of ordered partitions of $[2m]$ we could have used any $2m$-element set, and it is clear that the construction is functorial.

Here's a more self-contained description of this module. For simplicity, I'll consider only the case $n=2m$. Consider the vector space $V$ spanned by the set $P$ of ordered partitions of $[2m]$ into $m$ blocks of size two. The symmetric group $S_{2m}$ acts naturally on $V$. Now let $T$ be the set of ordered partitions of $[2m]$ with $m-2$ blocks of size two and one block of size 4, and for $t\in T$, let $\alpha(t)$ be the sum in $V$ of all the elements of $P$ from which $t$ be obtained by merging two adjacent blocks. The the quotient of $V$ by the $S_n$-module spanned by the $\alpha(t)$ is the desired $S_{2m}$-module, with dimension $E_{2m}$.

For example, with $n=4$, there are 6 ordered partitions:

12 | 34, 13 | 24, 14 | 23, 34 | 12, 24 | 13, and 23 | 14.

There is only one element $t\in T$, and $\alpha(t)$ is the sum of all six ordered partitions in $P$, so the quotient module has dimension $5=A_4$.

Instead of ordered partitions of $[2m]$ we could have used any $2m$-element set, and it is clear that the construction is functorial.

Here's a more self-contained description of this module. For simplicity, I'll consider only the case $n=2m$. Consider the vector space $V$ spanned by the set $P$ of ordered partitions of $[2m]$ into $m$ blocks of size two. The symmetric group $S_{2m}$ acts naturally on $V$. Now let $T$ be the set of ordered partitions of $[2m]$ with $m-2$ blocks of size two and one block of size 4, and for $t\in T$, let $\alpha(t)$ be the sum in $V$ of all the elements of $P$ from which $t$ can be obtained by merging two adjacent blocks. Then the quotient of $V$ by the $S_n$-module spanned by the $\alpha(t)$ is the desired $S_{2m}$-module, with dimension $A_{2m}$.

For example, with $n=4$, there are 6 ordered partitions:

12 | 34, 13 | 24, 14 | 23, 34 | 12, 24 | 13, and 23 | 14.

There is only one element $t\in T$, and $\alpha(t)$ is the sum of all six ordered partitions in $P$, so the quotient module has dimension $5=A_4$.

Instead of ordered partitions of $[2m]$ we could have used any $2m$-element set, and it is clear that the construction is functorial.

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created Mar 26, 2014 at 0:14

Ira Gessel

Here's a more self-contained description of this module. For simplicity, I'll consider only the case $n=2m$. Consider the vector space $V$ spanned by the set $P$ of ordered partitions of $[2m]$ into $m$ blocks of size two. The symmetric group $S_{2m}$ acts naturally on $V$. Now let $T$ be the set of ordered partitions of $[2m]$ with $m-2$ blocks of size two and one block of size 4, and for $t\in T$, let $\alpha(t)$ be the sum in $V$ of all the elements of $P$ from which $t$ be obtained by merging two adjacent blocks. The the quotient of $V$ by the $S_n$-module spanned by the $\alpha(t)$ is the desired $S_{2m}$-module, with dimension $E_{2m}$.

For example, with $n=4$, there are 6 ordered partitions:

12 | 34, 13 | 24, 14 | 23, 34 | 12, 24 | 13, and 23 | 14.

There is only one element $t\in T$, and $\alpha(t)$ is the sum of all six ordered partitions in $P$, so the quotient module has dimension $5=A_4$.

Instead of ordered partitions of $[2m]$ we could have used any $2m$-element set, and it is clear that the construction is functorial.