A canonical bijection from linear independent vectors to parking functions

Question

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.

Answer 1

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.

Answer 2

See "A braid group action on parking functions" by Gorsky and Gorsky.