Call an $n$-vector $v$ in $\mathbb{Z}^n$ cool when it has only entries 0 or 1 and the ones appear in only one block. Thus there are $n(n+1)/2$ such vectors. For $n=3$ they are:


[ <[ 1, 0, 0 ]>, <[ 1, 1, 0 ]>, <[ 0, 1, 0 ]>, <[ 1, 1, 1 ]>, <[ 0, 1, 1 ]>, <[ 0, 0, 1 ]> ].


Let $X\_n$ be the set of cool $n$-vectors. Call a subset $U \subset X\_n$ cool when $U$ has $n$ elements that are linearly independent.
There should be $(n+1)^{n-1}$ cool subsets of $X\_n$. For $n=3$ they are:


[ [ <[ 1, 0, 0 ]>, <[ 1, 1, 0 ]>, <[ 1, 1, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 1, 1, 0 ]>, <[ 0, 1, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 1, 1, 0 ]>, <[ 0, 0, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 0, 1, 0 ]>, <[ 1, 1, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 0, 1, 0 ]>, <[ 0, 1, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 0, 1, 0 ]>, <[ 0, 0, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 1, 1, 1 ]>, <[ 0, 0, 1 ]> ],


[ <[ 1, 0, 0 ]>, <[ 0, 1, 1 ]>, <[ 0, 0, 1 ]> ],


[ <[ 1, 1, 0 ]>, <[ 0, 1, 0 ]>, <[ 1, 1, 1 ]> ],


[ <[ 1, 1, 0 ]>, <[ 0, 1, 0 ]>, <[ 0, 1, 1 ]> ],


[ <[ 1, 1, 0 ]>, <[ 0, 1, 0 ]>, <[ 0, 0, 1 ]> ],


[ <[ 1, 1, 0 ]>, <[ 1, 1, 1 ]>, <[ 0, 1, 1 ]> ],


[ <[ 1, 1, 0 ]>, <[ 0, 1, 1 ]>, <[ 0, 0, 1 ]> ],


[ <[ 0, 1, 0 ]>, <[ 1, 1, 1 ]>, <[ 0, 1, 1 ]> ],


[ <[ 0, 1, 0 ]>, <[ 1, 1, 1 ]>, <[ 0, 0, 1 ]> ],


[ <[ 1, 1, 1 ]>, <[ 0, 1, 1 ]>, <[ 0, 0, 1 ]> ] ]



> 
> Question: Is there a canonical bijection from cool subsets of $X\_n$ to parking functions (that are counted by the same number $(n+1)^{n-1}$)?
> 
> 
> 


Background: The cool vectors correspond to the indecomposable representations of the $A\_n$-quiver algebra $A$ and the cool subsets to the bases of the Grothendieck group $K\_0(A)$ of $A$. I'm interested in a "canonical" bijection to parking functions to enter some statistics from homological algebra into findstat : findstat.org that has several statistics and maps for parking functions.
I can not really say what canonical means but it should behave nice under some standard statistics from homological algebra.
For example for such a canonical bijection, the number of simple vectors (those having only one non-zero entry) or the number of projective vectors (those having the last entry nonzero) in U should probably correspond to something nice for parking functions.



A canonical bijection from linear independent vectors to parking functions
They are in canonical bijection with the spanning trees of the complete graph $K\_{n+1}$ (for which the bijections with parking functions are well known).


Indeed, let $K\_{n+1}$ be the complete graph on the ground set $\{0,1,\ldots,n\}$. Denote $f\_0=0$ and consider $n$ linearly independent vectors $f\_1,\ldots,f\_n$. Denote further $e\_j=f\_j-f\_{j-1}$ for $j=1,\ldots,n$. They form another basis of the same $n$-dimensional space $W$ as $f\_j$'s. For an edge $\epsilon=ij$, $i<j$, of $K\_{n+1}$ we consider the vector $w(\epsilon)=f\_j-f\_i=e\_{i+1}+\ldots+e\_j$. Note that $n$ edges $w(\epsilon\_1),\ldots,w(\epsilon\_k)$ are linearly independent if and only if the set of edges $\epsilon\_i$'s does not contain cycles. Thus the bases of $W$ correspond to spanning trees of $K\_{n+1}$.


The above construction is a standard vector representation of the circuit matroid.


