Bubble Lattices II: Combinatorics

TL;DR

This work develops a triangular family of refinements for shuffle- and bubble-lattice combinatorics through three intertwined polynomials: the $F$-triangle, $H$-triangle, and $M$-triangle. It introduces two noncrossing simplicial complexes, $\Gamma(m,n)$ (noncrossing matchings) and $\Delta(m,n)$ (noncrossing bipartite), and shows they realize canonical join structures of semidistributive lattices, yielding shellability and precise topological types (pure, thin spheres with wedge decompositions). The authors derive explicit formulas for refined face counts, relate $F$- and $H$-triangles by substitutions, and connect these to the $M$-triangle of the shuffle lattice, including a conjectured closed form and a Delannoy-path interpretation of the positive parts. The results illuminate deep connections between shuffle/bubble lattices and classical Coxeter-Catalan phenomena, providing concrete enumerative tools and geometric interpretations that mirror Cambrian/noncrossing partition structures. Overall, the paper unifies lattice-theoretic, simplicial-topological, and enumerative perspectives to broaden understanding of bubble/shuffle analogs in combinatorics.

Abstract

We introduce two simplicial complexes, the noncrossing matching complex and the noncrossing bipartite complex. Both complexes are intimately related to the bubble lattice introduced in our earlier article "Bubble Lattices I: Structure" (arXiv:2202.02874). We study these complexes from both an enumerative and a geometric point of view. In particular, we prove that these complexes are shellable and give explicit formulas for certain refined face numbers. Lastly, we conjecture an intriguing connection of these refined face numbers to the so-called M-triangle of the shuffle lattice.

Figures (9)

• Figure 1: Two posets of shuffle words.
• Figure 2: Illustration of Proposition \ref{['prop:ncm_faces']}. In the vertices of $\Gamma(2,1)$, the two red nodes on the bottom correspond to $x_{1}$ and $x_{2}$ (from left to right), while the top blue node corresponds to $y_{1}$. The gray triangle represents a two-dimensional face.
• Figure 3: The graph of the noncrossing matching complex $\Gamma(2,2)$. Since $\Gamma(2,2)$ is flag, this graph determines the whole complex.
• Figure 4: The $22$$2$-Delannoy paths in the $2\times 2$-square.
• Figure 5: Illustrating the bijection from $q$-Delannoy paths to $(q+1)$-flags in $\Gamma^{+}$.

Theorems & Definitions (103)

• Theorem 1.1
• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: mcconville.muehle:bubbleI*Lemma 3.1
• Lemma 2.2: mcconville.muehle:bubbleI*Lemma 3.4
• Lemma 2.3: mcconville.muehle:bubbleI*Lemma 3.5
• Lemma 2.4: mcconville.muehle:bubbleI*Lemma 3.6
• Proposition 2.5: ziegler:polytopes*Theorem 8.19
• Theorem 2.6: bjorner:combinatorial*Theorem 1.3