Zig-Zag Posets: Theory & Applications

• Zig-zag posets are finite partially ordered sets defined by alternating cover relations, naturally corresponding to alternating permutations and graded chain structures.
• Their structure facilitates rigorous combinatorial analysis, exemplified by their role in Ehrhart theory, W-polynomials, and unimodal generating functions.
• These posets intersect with schemoid algebras and polyhedral geometry, offering applications in permutation statistics, magic labelings, and even chemical graph theory.

A zig-zag poset is a finite partially ordered set (poset) characterized by alternating cover relations, typically of the form $z_1 < z_2 > z_3 < z_4 > \ldots$ along a chain of elements. Zig-zag posets serve as a central object in enumerative combinatorics, polyhedral geometry, and algebraic combinatorics, connecting permutation statistics (especially alternating permutations), polytopal invariants, schemoids, and generalized Eulerian polynomials. They feature prominently in the paper of W-polynomials, Ehrhart theory of polytopes, and the gamma-nonnegativity and unimodality phenomena of poset-associated generating functions.

1. Definition and Structural Properties

The classical zig-zag poset %%%%1%%%% on $n$ elements $\{z_1, z_2, \ldots, z_n\}$ is defined by the cover relations:

• $z_1 < z_2$,
• $z_2 > z_3$,
• $z_3 < z_4$,
• $z_4 > z_5$, $\ldots$

This alternation endows $Z_n$ with a graded structure of rank $1$, in which all maximal chains have the same length. Zig-zag posets are naturally labeled and their linear extensions are in bijection with alternating (up-down) permutations. The canonical small category derived from a poset translates this order structure into categorical morphisms: objects are elements of $P$, and for $x \leq y$, there exists exactly one morphism from $x$ to $y$ [(Numata, 2016)].

The extended construction can include additional structure, such as a rank function $\rho$ and a join operation $x \vee y$, where, for $y = x \vee d$ with $\rho(y) = \rho(x) + \rho(d)$, morphisms $f_{x,y}^{(d)} : x \to y$ are introduced and compose additively via the join. Zig-zag behavior manifests in such categories as oscillations in composable join paths and difference operations.

2. Schemoids From Zig-Zag Posets

Schemoids are categorical generalizations of association schemes, aiming to encode combinatorial regularity in the structure of a small category. For a poset $P$, the canonical small category $C_P$ has morphisms $(y,x)$ for $x \leq y$. Morphism labeling via a difference operation $\delta : \{(y, x) \in P^2 | x \leq y\} \to P$ sets up a partition analogous to color classes in association schemes. The triple $(C_P, P, \delta)$ forms a schemoid under specific consistency conditions, such as the existence of bijections $\varphi_{x,y}$ between intervals in $P$, satisfying $\delta(o, \varphi_{x,y}(z)) = \delta(x, z)$ [(Numata, 2016)].

For zig-zag posets, especially when interpreted as ranked posets with specific join properties, these join and difference operations encode the characteristic alternation. Schemoid algebras constructed from these categories generalize the Bose–Mesner algebra, with algebraic invariants capturing intersection numbers and combinatorial symmetries.

3. Ehrhart Theory: Chain and Order Polytopes

The chain polytope $C(P)$ and the order polytope $\mathcal{O}(P)$ associated to a poset $P$ are convex polytopes in $\mathbb{R}^n$ defined by inequalities reflecting the poset relations. For zig-zag posets, the set of inequalities alternates, encoding their combinatorial structure. Stanley's foundational result shows that the Ehrhart $\delta$-polynomial of $C(P)$ is identical to the W-polynomial of $P$: $\delta(C(P); t) = W(P; t)$ [(Chen et al., 2016)].

For the zig-zag poset $Z_n$, the unimodality of the Ehrhart $\delta$-polynomial of $C(Z_n)$ was established. This result confirms Kirillov’s conjecture, situating the $\delta$-polynomial within the framework of naturally labeled graded poset W-polynomials, whose unimodality follows from Gasharov's theorem for rank $1$ or $2$ [(Chen et al., 2016)]. The implication is a symmetric and "well-behaved" enumeration of lattice points in dilates of the chain polytope, enabling connections to Kostka numbers and partition functions.

4. Permutation Statistics, Swap Statistic, and h*-Polynomials

Alternating permutations index the linear extensions of the zig-zag poset, giving rise to the Euler zig-zag numbers. The canonical unimodular triangulation of the order polytope $\mathcal{O}(Z_n)$ yields simplices parametrized by these permutations [(Coons et al., 2019)]. The shelling process induces a combinatorial interpretation for the h*-polynomial of $\mathcal{O}(Z_n)$: each simplex, corresponding to permutation $\sigma$, is glued along exactly $\text{swap}(\sigma)$ facets, where $\text{swap}(\sigma)$ counts positions $i$ such that swapping $i$ and $i+1$ produces another alternating permutation.

$$h^*_{\mathcal{O}(Z_n)}(t) = \sum_{\sigma \in A_n} t^{\text{swap}(\sigma)}$$

This correspondence conveys geometric and combinatorial information about the polytope and the poset and has implications for symmetric and unimodal distribution of swap numbers. The swap statistic shares a bijection with rank-selected chains of order ideals, linking permutation statistics with flag h-vectors in lattice structures.

5. Zig-Zag Eulerian Polynomials and Gamma-Nonnegativity

For any poset $P$, the $P$-Eulerian polynomial is the descent generating function over linear extensions of $P$, closely related to the order polynomial via $P$-partition theory. Zig-zag Eulerian polynomials $Z_n(t)$, associated to naturally labeled zig-zag posets, are proven to be gamma-nonnegative, thus have symmetric and unimodal coefficients [(Petersen et al., 11 Mar 2024)]. The gamma-nonnegative expansion is explicitly

$$Z_n^r(t) = t \cdot \sum_{0 \leq 2j \leq n-r-1} \gamma_{n,j} t^j (1+t)^{n - r - 1 - 2j}$$

with $\gamma_{n,j} \geq 0$.

A plausible implication is conjectured real-rootedness for $Z_n(t)$ and its $r$-generalizations, which, if validated, would strengthen log-concavity and combinatorial regularity. These polynomials encode the distribution of "big returns" over alternating permutations and provide a unifying lens on statistics such as Entringer numbers, Narayana polynomials, and Catalan structures.

6. Combinatorial and Geometric Interpretations

The theory discerns rich connections between zig-zag posets and a variety of combinatorial and geometric objects:

• Both order and chain polytopes of $Z_n$ have identical Ehrhart polynomials (up to a shift), with their $h^*$-polynomial given by $Z_n(t)/t$ [(Petersen et al., 11 Mar 2024)].
• Magic labelings of path graphs and enumerations of Kekulé structures in chemistry (perfect matchings of benzenoid hydrocarbons) can be equivalently counted by the order polynomial values $\Omega_n(m+1)$.
• Decreasing binary trees and Jacobi permutations are in bijection with alternating permutations possessing specific statistics (e.g., big returns), providing yet another interpretation for $Z_n(t)$.
• The h-polynomial of the simplicial complex induced by a Coxeter cone of a poset equals $A_P(t)/t$.

This multifaceted viewpoint consolidates different domains—combinatorics, geometry, chemistry—demonstrating the unifying power of zig-zag poset theory.

7. Open Problems and Research Directions

Several conjectures and open questions frame ongoing research:

• Are zig-zag Eulerian polynomials $Z_n(t)$ and their generalizations real-rooted for all $n$ and $r$? Does the coefficient sequence always become log-concave?
• Can the $\gamma$-vector satisfy the Frankl–Füredi–Kalai inequalities, indicating it is the $f$-vector of a balanced simplicial complex?
• Is there a direct combinatorial explanation for the symmetric and unimodal distribution of big returns on alternating permutations?
• What are the linear recurrence relations for refined Entringer-number polynomials $U_{n,r}(s,t)$, and how do these interact with big return statistics and alternation?

Continued investigation in the literature, including computational explorations and extension to refined statistics, motivates further research at the intersection of algebraic combinatorics, geometric enumeration, and applied disciplines.

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Zig-zag posets thus crystallize essential combinatorial phenomena: their deployment in the construction of schemoid algebras, explorations of unimodality and symmetric polynomial expansions, and combinatorial-geometric equivalences offer a robust framework for both theoretical inquiry and application across mathematics and related fields.