Primeness of generalized parking functions

Abstract

Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a direct sum of prime ones. In this article, we extend the notion of primeness to three generalizations of classical parking functions: vector parking functions, $(p,q)$-parking functions, and two-dimensional vector parking functions. We study their enumeration by obtaining explicit formulas for the number of prime vector parking functions when the vector is an arithmetic progression, prime $(p,q)$-parking functions, and prime two-dimensional vector parking functions when the weight matrix is an affine transformation of the coordinates.

Figures (6)

• Figure 1: (a) The increasing parking function $\boldsymbol{a}=(0,0,2,3)$ and parking function $\boldsymbol{b}=(2,0,3,0)$ correspond to the same Catalan path $L_{\boldsymbol{a}}=L_{\boldsymbol{b}}$. (b) The parking function $\boldsymbol{b}$ is represented by adding labels $j\in\mathbb{N}_4$ to the vertical edges of $L_{\boldsymbol{b}}$ such that $j$ has $x$-coordinate $b_{j}$.
• Figure 2: The parking function $\boldsymbol{a}=(0,3,1,0)$ can be decomposed as $\boldsymbol{a}=\boldsymbol{a_1}\oplus\boldsymbol{a_2}$ with $\boldsymbol{a_1}=(0,1,0)$ and $\boldsymbol{a_2}=(0)$. Note that $\boldsymbol{b_1}=\boldsymbol{a_1}=(0,1,0)$ and $\boldsymbol{b_2}=\boldsymbol{a_2}+3=(3)$, and $\boldsymbol{a}$ is a shuffle of $\boldsymbol{b_1}$ and $\boldsymbol{b_2}$. Here, $B_1=\{0,2, 3\}$ and $B_2=\{1\}$, where $B_i$ is the set of positions of $\boldsymbol{b_i}$ in $\boldsymbol{a}$, for $i=1, 2$.
• Figure 3: (a) For $\boldsymbol{u}=(1,1,3)$, the lattice path $L_{\boldsymbol{u}}=ENNEEN$ has vertical edges given by $\boldsymbol{u}$. (b) For $\boldsymbol{a}=(0,2,0)$, the labeled lattice path $L_{\boldsymbol{a}}=NNEENE$ has right boundary $\boldsymbol{u}$, confirming that $\boldsymbol{a}$ is a $\boldsymbol{u}$-parking function.
• Figure 4: (a) For the pair $\boldsymbol{a}=(3,0,3)$ and $\boldsymbol{b}=(1,0,1,0)$, $L_{\boldsymbol{b}}=NNENNEE$ is weakly above $L^\perp_{\boldsymbol{a}}=ENNNEEN$, confirming that $(\boldsymbol{a},\boldsymbol{b})$ is a $(3,4)$-parking function. (b) For the pair $\boldsymbol{a'}=(0,3,3)$ and $\boldsymbol{b'}=(0,0,2,2)$, $L_{\boldsymbol{b'}}=NNEENNE$ is not weakly above $L^{\perp}_{\boldsymbol{a'}}=ENNNEEN$, and thus $(\boldsymbol{a'},\boldsymbol{b'})$ is not a (3,4)-parking function.
• Figure 5: (a) For $\boldsymbol{U}=\{z_{k,\ell}=(u_{k,\ell},v_{k,\ell}):(0,0)\leq(k,\ell)\leq(3,4)\}$, we show the edge weights on the directed graph $G_{3,4}(\boldsymbol{U})$. Subsequent figures will omit the arrows. (b) We show the path $P=NNEENEN\in\mathcal{L}(3,4)$ and the weights on the edges of $G_{3,4}(\boldsymbol{U})$ for $\boldsymbol{U}$ from \ref{['ex:two-dim']}. The edge weights on $P$ are $(\mathrm{wt}(e_1),\ldots,\mathrm{wt}(e_7))=(1,1,3,3,3,4,4)$, corresponding to the weight sequences $(3,3,4)$ and $(1,1,3,4)$ when restricted to the horizontal and vertical edges, respectively.

Theorems & Definitions (46)

• Definition 1.1
• Lemma 1.2
• proof
• Example 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4: SY22
• Lemma 2.5
• proof
• Example 2.6