Orbitmesy in Combinatorial Dynamics

• Orbitmesy is a phenomenon in dynamical algebraic combinatorics where selected orbits average a statistic that equals the global average over the entire set.
• Promotion on self-dual zig-zag posets is employed to generate and classify orbitmesic orbits using techniques such as swap maps and pattern avoidance.
• Explicit formulas for orbit sizes and invariant statistics like the antipodal and total sum provide actionable insights into combinatorial dynamics.

Orbitmesy refers to a phenomenon in dynamical algebraic combinatorics generalizing homomesy, in which only particular orbits of a combinatorial action exhibit averaging properties for certain statistics. Specifically, a subset (orbit) of a set acted on by a bijection is said to be orbitmesic with respect to a statistic if the orbit's average of that statistic matches the global average over the entire set. This concept, introduced alongside a comprehensive investigation of promotion on self-dual posets, particularly zig-zag (fence) posets, provides new perspectives on the distribution of invariant values in combinatorial dynamical systems (Banaian et al., 26 Aug 2025).

1. Mathematical Definition and Connection to Homomesy

Orbitmesy is formulated as follows. Let $X$ be a finite set, $\phi : X \to X$ a bijective action, and $\eta : X \to \mathbb{R}$ a statistic. An orbit $\mathcal{O}$ under $\phi$ is said to be orbitmesic (with respect to $\eta$) precisely when

$$\frac{1}{|\mathcal{O}|}\sum_{x \in \mathcal{O}} \eta(x) = \frac{1}{|X|}\sum_{x \in X} \eta(x).$$

If this property holds for every orbit, the statistic is homomesic with respect to $\phi$. Orbitmesy thus captures a “partial homomesy” situation in which only certain orbits exhibit the desired averaging, while others do not align with the global average.

In dynamical algebraic combinatorics, homomesy has become a central phenomenon for various combinatorial actions and statistics. Orbitmesy expands this framework, allowing the rigorous identification of “nice” orbits where averaging properties persist but not universally across all orbits.

2. Promotion Action on Increasing Labelings of Self-Dual Posets

A principal focus of the orbitmesy theory is the promotion action on increasing labelings of specific posets, such as zig-zag posets (also called fence posets). These posets are sequences of elements ordered with alternating cover relations. The set $Z_n$ denotes a zig-zag poset on $n$ elements. An increasing labeling $f : P \to [q]$ assigns integers $1,\dots,q$ to poset elements so that $x < y \implies f(x) < f(y)$.

Promotion in this context is realized via combinatorial operations analogous to jeu de taquin slides, generating orbits in the space $\Inc^q(Z_n)$ of increasing labelings. The analysis concentrates on $Z_4$, the four-element zig-zag poset, providing explicit orbitmesy classifications for two statistics: the antipodal sum and the total sum.

3. Statistics Exhibiting Orbitmesy: Antipodal Sum and Total Sum

Two central statistics are considered:

• Antipodal sum: For a self-dual poset $P$ with an order-reversing involution $\kappa : P \to P$, the antipodal sum at $x$ is defined as

$A_x(f) = f(x) + f(\kappa(x)).$

In zig-zag posets, this statistic naturally distinguishes between exterior (leftmost and rightmost) and interior (middle) element pairs.

• Total sum: The total sum statistic is simply

$Tot(f) = \sum_{x \in P} f(x).$

The swap operation, $swap(f)(x) = q+1 - f(\kappa(x))$, is shown to satisfy the property $A_x(f) + A_x(swap(f)) = 2(q+1)$ for any $x$, demonstrating that swap-orbits are always homomesic with respect to the antipodal sum statistic. Moreover, swap commutes or anti-commutes with promotion, providing a structural route to identifying orbitmesic orbits.

4. Classification of Orbitmesic Promotion Orbits and General Results

For the zig-zag poset $Z_4$, the orbitmesic orbits under promotion are fully classified:

• A promotion orbit in $\Inc^q(Z_4)$ is orbitmesic for the antipodal sum statistic if and only if it either avoids the subword pattern “1324” in label read-off, or satisfies a “balanced” property where the gap between the first two entries equals the gap between the last two, modulo $q$ (see Definition 6.5 of (Banaian et al., 26 Aug 2025)).
• If an orbit contains the pattern “1324,” it exhibits orbitmesy only when the balanced condition is met.
• For the total sum statistic, all promotion orbits in $\Inc^q(Z_4)$ are $2(q+1)$-mesic, i.e., orbit averages for $Tot$ always equal $2(q+1)$.

Theoretical results (Theorem 4.6, Theorem 5.7) establish criteria for orbitmesy and provide explicit formulas for promotion orbit sizes, such as

$$|\mathcal{O}(f)| = \frac{\tau\,\ell}{\gcd(r\ell/q,\,\tau)},$$

where $\tau$ is the size of the packed (deflated) orbit, $\ell$ is the binary content word period, and $r$ counts distinct labels in $f$.

The swap map is essential: if a promotion orbit is swap-closed (i.e., contains $swap(f)$ for every $f$), it is necessarily orbitmesic for the antipodal sum and total sum statistics. This construction enables the identification of infinite families of orbitmesic orbits in any self-dual poset.

5. Combinatorial Techniques: Deflation, Packed Labelings, and Infinite Liftings

Orbitmesy classification is supported by combinatorial reduction processes:

• Deflation: Labelings are “deflated” by mapping each used label to a contiguous segment ($1,2,\dots,r$), facilitating orbit analysis in the packed subset. Orbit properties such as period and swap-closure can then be lifted from the packed setting to the full label space.
• Packed labelings: These are increasing labelings using consecutive numbers, forming a finite model where promotion orbit sizes and properties are more tractable.
• Pattern avoidance and balance: Small patterns or balance conditions in the labeling (e.g., avoiding “1324”) serve as diagnostic tools for orbitmesy status.

Infinite families of orbitmesic orbits are constructed by combining packed analysis, swap-closure, and properties preserved under deflation.

6. Implications, Extensions, and Connections

Orbitmesy refines the understanding of “statistic averaging” in combinatorial dynamical systems. While global homomesy may fail, large structured subsets of orbits (often characterized by swap-closure or pattern properties) retain global averages for key statistics. This demonstrates that invariant phenomena in promotion dynamics on self-dual posets extend beyond the reach of classical homomesy.

Moreover, the conceptual and technical tools introduced for orbitmesy—explicit orbit size formulas, swap-closure, pattern-based criteria—open pathways for further studies in combinatorics and related dynamical systems. The framework allows for systematic enumeration and control of the orbit-averaging properties, with plausible implications for symmetry and structural analysis in other settings.

7. Summary of Key Formulas and Theorems

Concept	Formula/Definition	Notes
Orbitmesy criterion	$\frac{1}{|\mathcal{O}|}\sum_{f \in \mathcal{O}} \eta(f) = \frac{1}{|\Inc^q(P)|}\sum_{f \in \Inc^q(P)} \eta(f)$	Holds for specified orbits
Promotion orbit size	$$|\mathcal{O}(f)| = \frac{\tau\,\ell}{\gcd(r\ell/q,\,\tau)}$$	See Theorem 4.6
Swap map	$swap(f)(x) = q+1 - f(\kappa(x))$	Swap-closure implies orbitmesy
Antipodal sum	$A_x(f) = f(x) + f(\kappa(x))$	Statistic under consideration
Balanced labeling	Gap equivalence in entries—see Definition 6.5	Restricts orbitmesy class

These formulas encode the essential criteria for orbitmesy in the combinatorial framework, underpinning both theoretical completeness and practical enumeration of orbitmesic structures.

In conclusion, orbitmesy provides a nuanced generalization of homomesy in dynamical algebraic combinatorics, enabling the identification, classification, and infinite generation of “averaging” orbits for key statistics under combinatorial actions such as promotion on self-dual posets (Banaian et al., 26 Aug 2025).