[Paper Review] A Fuss-Catalan variation of the caracol flow polytope

Martha Yip|arXiv (Cornell University)|Oct 22, 2019

Advanced Combinatorial Mathematics9 references4 citations

TL;DR

This paper introduces a refined combinatorial model based on truncated unified diagrams and $k$-parking numbers to compute the volume of flow polytopes associated with $k$-caracol graphs—non-planar generalizations of the classical caracol and Pitman–Stanley graphs. It establishes a Fuss-Catalan-type volume formula involving rational Catalan numbers and powers of $k$, and proves a combinatorial correspondence between in-degree and out-degree gravity diagrams, revealing a deep structural symmetry in related polytopes.

ABSTRACT

Recently, a combinatorial interpretation of Baldoni and Vergne's generalized Lidskii formula for the volume of a flow polytope was developed by Benedetti et al.. This converts the problem of computing Kostant partition functions into a problem of enumerating a set of objects called unified diagrams. We devise an enhanced version of this combinatorial model to compute the volumes of flow polytopes defined on a family of graphs called the k-caracol graphs, resulting in the first application of the model to non-planar graphs. At k=1 and k=n-1, we recover results for the classical caracol graph and the Pitman--Stanley graph. Furthermore, we introduce the notion of in-degree gravity diagrams for flow polytopes, which are equinumerous with (out-degree) gravity diagrams considered by Benedetti et al.. We show that for the k-caracol flow polytopes, these two kinds of gravity diagrams satisfy a natural combinatorial correspondence, which raises an intriguing question on the relationship in the geometry of two related polytopes.

Motivation & Objective

• To extend the unified diagram model to non-planar $k$-caracol graphs, enabling volume computation without constant term identities.
• To generalize classical results on caracol and Pitman–Stanley polytopes to a broader family of graphs parameterized by $k$.
• To introduce and combinatorially interpret $k$-parking numbers as a bridge between generalized Fuss-Catalan numbers and parking functions.
• To establish a natural bijection between in-degree and out-degree gravity diagrams for $k$-caracol polytopes, revealing a geometric duality.
• To provide a combinatorial explanation for the $k^{k(n-k)-2}$ factor in volume formulas, previously undetected in the $k=1$ case.

Proposed method

• Introduce truncated unified diagrams as a refinement of the unified diagram model to capture the $k^{k(n-k)-2}$ factor in volume formulas.
• Define $k$-parking numbers $T_k(r,i) = (r+1)^{i-1} \binom{k(r+1)}{r-i}$, which enumerate completions of truncated diagrams and interpolate between Fuss-Catalan numbers and parking functions.
• Construct a bijection $\Xi$ between out-degree gravity diagrams of $k$-caracol and $k$-multicaracol graphs, proving volume equivalence at certain net flows.
• Apply a binomial transform to $k$-parking numbers to derive the volume formula for the $k$-caracol polytope with net flow $(1,\ldots,1,-n)$.
• Use the generalized Lidskii formula and unified diagram enumeration to compute normalized volumes of flow polytopes without relying on constant term identities.
• Establish a $k$-to-$1$ correspondence between unified diagrams of $\operatorname{MCar}_{n-k+2}^{(k)}$ and $\operatorname{Car}_{n+1}^{(k)}$ at specific net flows, suggesting a geometric map.

Experimental results

Research questions

• RQ1 How can the unified diagram model be extended to non-planar graphs such as the $k$-caracol?
• RQ2 What combinatorial object explains the $k^{k(n-k)-2}$ factor in the volume formula for $k$-caracol polytopes?
• RQ3 Is there a natural combinatorial correspondence between in-degree and out-degree gravity diagrams for $k$-caracol polytopes?
• RQ4 Can the volume of the $k$-caracol polytope with net flow $(1,\ldots,1,-n)$ be expressed in terms of generalized Fuss-Catalan numbers and $k$-parking numbers?
• RQ5 What is the geometric significance of the $k$-to-$1$ map between unified diagrams of $\operatorname{MCar}_{n-k+2}^{(k)}$ and $\operatorname{Car}_{n+1}^{(k)}$?

Key findings

• The volume of the $k$-caracol polytope with net flow $(1,\ldots,1,-n)$ is $\operatorname{Cat}(n-k,k(n-k)-1) \cdot k^{k(n-k)-2} \cdot n^{n-k-1}$, a Fuss-Catalan-type formula involving rational Catalan numbers.
• The $k$-parking numbers $T_k(r,i) = (r+1)^{i-1} \binom{k(r+1)}{r-i}$ provide a combinatorial interpretation of the $k^{k(n-k)-2}$ factor via a vehicle-parking scenario.
• A natural bijection $\Xi$ exists between out-degree gravity diagrams of $\operatorname{Car}_{n+1}^{(k)}$ and $\operatorname{MCar}_{n-k+2}^{(k)}$, proving volume equivalence at net flow $(kx,y^{n-k},-kx-(n-k)y)$.
• The in-degree gravity diagrams for $k$-caracol polytopes are equinumerous with out-degree gravity diagrams, and a combinatorial correspondence between them is established.
• The volume of the $k$-multicaracol polytope with net flow $(kx,y^{n-k},-kx-(n-k)y)$ is $\operatorname{Cat}(a,ka-1) \cdot (kx)^{ka-1} (kx+ay)^{a-1}$, where $a = n-k$.
• The formula for the $k$-caracol polytope volume is derived via a binomial transform of $k$-parking numbers, with the power of $k$ arising from completion counts of truncated unified diagrams.