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Let $V$ be a vector space. Let us say that a finite set $X$ of vectors in $V$ is harmonic if for $B \subseteq X$, $$ B \text{ is a basis of } V \implies X \setminus B \text{ is a basis of }V. $$ Let us say that the basis number of a harmonic set $X$ is the number of subsets of $X$ that are bases of $V$.

Which integers arise as the basis number of some harmonic set?

Clearly, the number of bases is even (unless $X$ is empty). Can it be an arbitrary even number? Can we get, e.g., 10 or 6?

This set of integers is closed with respect to multiplication and contains 2. But it consists of not only powers of 2. It is easy to obtain, e.g., $2n^2$ for any $n$ (take $2n$ vectors in an $n$-dimensional space: $n$ vectors from a hyperspace and $n$ from another hyperspace in general position).

Let $V$ be a vector space. Let us say that a finite set $X$ of vectors in $V$ is harmonic if for $B \subseteq X$, $$ B \text{ is a basis of } V \implies X \setminus B \text{ is a basis of }V. $$ Let us say that the basis number of a harmonic set $X$ is the number of subsets of $X$ that are bases of $V$.

Which integers arise as the basis number of some harmonic set?

Clearly, the number of bases is even (unless $X$ is empty). Can it be an arbitrary even number? Can we get, e.g., 10 or 6?

This set of integers is closed with respect to multiplication and contains 2. But it consists of not only powers of 2. It is easy to obtain, e.g., 18 (take 6 vectors in a 3-dimensional space: 3 vectors from a plane and 3 from another plane in general position).

Let us call a finite set of vectors harmonic if the complement of any of its basis is a basis too ($B\subseteq X$ is a basis $\Rightarrow$ $X\setminus B$ is a basis).

How many bases can a harmonic set have?

(A basis is a non-ordered basis in this question.)

Clearly, the number of bases is even (for a nonempty set). Can it be an arbitrary even number? Can we get, e.g., 10 or 6?

This set of integers is closed with respect to multiplication and contains 2. But it consists of not only powers of 2. It is easy to obtain, e.g., $2n^2$ for any $n$ (take $2n$ vectors in an $n$-dimensional space: $n$ vectors from a hyperspace and $n$ from another hyperspace in general position).

Let $V$ be a vector space. Let us say that a finite set $X$ of vectors in $V$ is harmonic if for $B \subseteq X$, $$ B \text{ is a basis of } V \implies X \setminus B \text{ is a basis of }V. $$ Let us say that the basis number of a harmonic set $X$ is the number of subsets of $X$ that are bases of $V$.

Which integers arise as the basis number of some harmonic set?

Clearly, the number of bases is even (unless $X$ is empty). Can it be an arbitrary even number? Can we get, e.g., 10 or 6?

This set of integers is closed with respect to multiplication and contains 2. But it consists of not only powers of 2. It is easy to obtain, e.g., $2n^2$ for any $n$ (take $2n$ vectors in an $n$-dimensional space: $n$ vectors from a hyperspace and $n$ from another hyperspace in general position).

Let us call a finite set of vectors harmonic if the compliment of any its basis is a basis too ($B\subseteq X$ is a basis $\Rightarrow$ $X\setminus B$ is a basis).

How many bases a harmonic set can have?

(A basis is a non-ordered basis in this question.)

Clearly, the number of bases is even (for a nonempty set). Can it be an arbitrary even number? Can we get, e.g., 10 or 6?

This set of integers is closed withe respect to multiplication and contains 2. But it consists of not only powers of 2. It is easy to obtain, e.g., $2n^2$ for any $n$ (take $2n$ vectors in an $n$-dimensional space: $n$ vectors from a hyperspace and $n$ from another hyperspace in general position).

Let us call a finite set of vectors harmonic if the complement of any of its basis is a basis too ($B\subseteq X$ is a basis $\Rightarrow$ $X\setminus B$ is a basis).

How many bases can a harmonic set have?

(A basis is a non-ordered basis in this question.)

Clearly, the number of bases is even (for a nonempty set). Can it be an arbitrary even number? Can we get, e.g., 10 or 6?

This set of integers is closed with respect to multiplication and contains 2. But it consists of not only powers of 2. It is easy to obtain, e.g., $2n^2$ for any $n$ (take $2n$ vectors in an $n$-dimensional space: $n$ vectors from a hyperspace and $n$ from another hyperspace in general position).