Return to Question

Let $M(n,k)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$. Note that the signed permutation group (the Coxeter group of type $B_n$) acts on this matroid. Questions:

1. Does this matroid have a name?
2. Has it been studied before?
3. Is there a nice formula for its characteristic polynomial?

Here are some boring special cases:

• $M(n,n)$ is the Booelan matroid on $2n$ elements.

• $M(n,n-1)$ is the uniform matroid of rank $2n-1$ on $2n$ elements.

• $M(n,0)$ is the direct sum of $n$ copies of the uniform matroid of rank 1 on 2 elements.

The first interesting case is $M(3,1)$, which has rank 4 and characteristic polynomial

$$q^4 - 6q^3 + 15q^2 - 17q + 7$$

I am also interested in truncations of this matroid. That is, let $M(n,k,d)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$ and has size at most $d$. All of the same questions apply!

Remark: I would like to regard these matroids as type B analogues of uniform matroids. Uniform matroids are the permutation-invariant matroids on the ground set $\{1,\ldots,n\}$, while these are the signed-permutation-invariant matroids on the ground set $\{\pm 1,\ldots,\pm n\}$.

Let $M(n,k)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$. Note that the signed permutation group (the Coxeter group of type $B_n$) acts on this matroid. Questions:

1. Does this matroid have a name?
2. Has it been studied before?
3. Is there a nice formula for its characteristic polynomial?

Here are some boring special cases:

• $M(n,n)$ is the Boolean matroid on $2n$ elements.

• $M(n,n-1)$ is the uniform matroid of rank $2n-1$ on $2n$ elements.

• $M(n,0)$ is the direct sum of $n$ copies of the uniform matroid of rank 1 on 2 elements.

The first interesting case is $M(3,1)$, which has rank 4 and characteristic polynomial

$$q^4 - 6q^3 + 15q^2 - 17q + 7$$

I am also interested in truncations of this matroid. That is, let $M(n,k,d)$ be the matroid on the ground set $\{\pm 1,\ldots,\pm n\}$ for which a set is independent if and only if it contains at most $k$ pairs $\pm i$ and has size at most $d$. All of the same questions apply!

Remark: I would like to regard these matroids as type B analogues of uniform matroids. Uniform matroids are the permutation-invariant matroids on the ground set $\{1,\ldots,n\}$, while these are the signed-permutation-invariant matroids on the ground set $\{\pm 1,\ldots,\pm n\}$.