An area-bounce exchanging bijection on a large subset of Dyck paths

TL;DR

This work tackles the long-standing problem of a bijection exchanging area and bounce on Dyck paths by constructing an explicit bijection that is natural on a large, exponentially rich subset of paths. The approach hinges on a calculus of path operators and AB-pairs, assembling a flip map $oldsymbol{ ext{Phi}}$ between two structured families and proving a symmetrical distribution $G_n(q,t)=G_n(t,q)$ on that subfamily. The authors further extend the construction via a two-parameter framework to broaden the AB-pair landscape and develop area-minimal and bounce-minimal theory, including operators $U_i$ and $D_i$ and results about intermediate ab-values. Finally, they connect the distinct sums $oldsymbol{a}( au)+oldsymbol{b}( au)$ to Johnson's $q$-Bell polynomial, showing the number of distinct sums equals the number of nonzero coefficients in $B_n(q)$, thereby linking Dyck-path statistics to a classical $q$-analogue and advancing Haglund's open problem with a concrete combinatorial bridge.

Abstract

It is a longstanding open problem to find a bijection exchanging area and bounce statistics on Dyck paths. We settle this problem for an exponentially large subset of Dyck paths via an explicit bijection. Moreover, we prove that this bijection is natural by showing that it maps what we call bounce-minimal paths to area-minimal paths. As a consequence of the proof ideas, we show combinatorially that a path with area $a$ and bounce $b$ exists if and only if a path with area $b$ and bounce $a$ exists. We finally show that the number of distinct values of the sum of the area and bounce statistics is the number of nonzero coefficients in Johnson's $q$-Bell polynomial.

Figures (14)

• Figure 1: The Dyck path, $\pi = \mathop{\mathrm{N}}\nolimits^3 \mathop{\mathrm{E}}\nolimits^2 \mathop{\mathrm{N}}\nolimits \mathop{\mathrm{E}}\nolimits \mathop{\mathrm{N}}\nolimits \mathop{\mathrm{E}}\nolimits^2 \mathop{\mathrm{N}}\nolimits^2 \mathop{\mathrm{E}}\nolimits^2$ has area sequence $a_\pi = (0,1,2,1,1,0,1)$, area $\mathbf{a}(\pi)=6$, bounce path (in red) $p_{7, (3,2,2)}$, bounce points $(0,3,5,7)$ and bounce $\mathbf{b}(\pi)=6$. This path has $1$ floating cell.
• Figure 2: Illustration of the action of $S_i$.
• Figure 3: An illustration of \ref{['prop:si_not_bottom']} where $\alpha_i > \alpha_{i+1}$, $k \ge 0$ and $\delta = \alpha_i - \alpha_{i+1}$.
• Figure 4: Examples of the action of $B_{i,k}$.
• Figure 5: The bounce and row maps annotated on $p_{13, \lambda}$, where $\lambda = (6, 3, 3, 1)$. Thus, $\overline{\lambda}_1=6$ and $\overline{\lambda}_2=3$. Note that $\mathcal{I}_{\mathop{\mathrm{bnc}}\nolimits}(\lambda) = \{(1, 1), (1, 2), (2, 1)\}$ and $\mathcal{I}_{\mathop{\mathrm{row}}\nolimits}(\lambda) = \{(1, 1), (1, 2), (1, 3), (2, 1)\}$

Theorems & Definitions (57)

• Definition 2.1
• Proposition 2.2: haglund-2003
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• Definition 3.5
• Definition 3.6