Loop Fence Poset in Combinatorics

• Loop fence poset is defined as the cyclic closure of a linear fence poset, embodying an alternating up–down pattern arranged in a circle.
• Its core enumerative invariant, the circular rank polynomial, exhibits symmetry and unimodality, reflecting the structure of lower ideals in distributive lattices.
• The framework connects combinatorial bijections, matrix-based formulations, rowmotion dynamics, and q-deformed Markov numbers, bridging theory and application.

A loop fence poset, also called a circular fence poset, is the circular analogue of a fence poset obtained by closing the alternating up–down pattern into a cycle. In the literature this object appears in several equivalent guises: as $F^\circ(a)$ obtained from a linear fence $F(a)$ by identifying $x_{n+1}=x_1$, as $F^c(\beta)$ defined directly by cyclic cover relations on $\{x_1,\dots,x_n\}$, and as $\overline F(\alpha)$ produced from an oriented fence by adjoining the relation $x_R\succ x_L$. Its basic enumerative invariant is the rank polynomial of the distributive lattice of lower ideals, denoted variously by $R^\circ(a;q)$, $R^c(q;\beta)$, or $\overline R(\alpha;q)$. The subject connects enumerative combinatorics, distributive lattices, rowmotion, and $F(a)$0-deformed Markov numbers (Oğuz et al., 2021, Elizalde et al., 2022, Oğuz, 2022).

1. Definition and notation

For a composition $F(a)$1 of $F(a)$2, the linear fence poset $F(a)$3 has ground set $F(a)$4 and cover relations determined by alternating “up-segments” and “down-segments.” Concretely, the $F(a)$5 part $F(a)$6 corresponds to a maximal chain of length $F(a)$7 in the usual zig–zag pattern. Its distributive lattice of lower ideals is

$F(a)$8

and the rank-generating polynomial is

$F(a)$9

This is the linear predecessor of the loop fence construction (Oğuz et al., 2021).

The circular version is defined when the composition has even length. One description starts from $x_{n+1}=x_1$0 on $x_{n+1}=x_1$1 vertices $x_{n+1}=x_1$2 and forms $x_{n+1}=x_1$3 by imposing the extra identification $x_{n+1}=x_1$4, obtaining a poset on exactly $x_{n+1}=x_1$5 vertices arranged around a cycle, with the same up–down pattern around the circle. Equivalently, one may define $x_{n+1}=x_1$6 for $x_{n+1}=x_1$7 on the ground set $x_{n+1}=x_1$8, with ascending internal covers on odd segments, descending internal covers on even segments, and a closing cover $x_{n+1}=x_1$9 (Oğuz et al., 2021, Elizalde et al., 2022).

For the circular fence, the lower-ideal lattice is

$F^c(\beta)$0

and the rank polynomial is

$F^c(\beta)$1

The different notations reflect different treatments of the same cyclic object (Oğuz et al., 2021, Elizalde et al., 2022).

2. Oriented posets and the $F^c(\beta)$2 rank-matrix formalism

A systematic construction of loop fences is given through oriented posets. An oriented poset is a finite poset $F^c(\beta)$3 equipped with distinguished elements $F^c(\beta)$4, written $F^c(\beta)$5. Its rank polynomial is

$F^c(\beta)$6

and this is refined according to whether $F^c(\beta)$7 or $F^c(\beta)$8 lie in the ideal. The four refined polynomials are packaged into a $F^c(\beta)$9 matrix

$\{x_1,\dots,x_n\}$0

The key structural fact is multiplicativity: if $\{x_1,\dots,x_n\}$1 and $\{x_1,\dots,x_n\}$2 are concatenated by adding a single cover-relation $\{x_1,\dots,x_n\}$3, then the new rank matrix is exactly the product of the two original matrices (Oğuz, 2022).

Loop closure is encoded by trace. If one adjoins the relation $\{x_1,\dots,x_n\}$4 to an oriented poset, then

$\{x_1,\dots,x_n\}$5

For loop fences, this gives a direct linear-algebraic model for the circular rank polynomial (Oğuz, 2022).

Chains provide the basic building blocks. The two oriented chains are

$\{x_1,\dots,x_n\}$6

Their matrices are

$\{x_1,\dots,x_n\}$7

where

$\{x_1,\dots,x_n\}$8

A fence poset $\{x_1,\dots,x_n\}$9 is then obtained by alternating concatenations of up- and down-chains, so its oriented-poset rank matrix is a corresponding product of these elementary matrices (Oğuz, 2022).

3. Rank polynomials, symmetry, and unimodality

The fundamental enumerative theorem for loop fences is rank symmetry. For every composition $\overline F(\alpha)$0 of even length, the circular-fence rank polynomial satisfies

$\overline F(\alpha)$1

so the coefficient sequence is palindromic (Oğuz et al., 2021). In the notation of $\overline F(\alpha)$2, if $\overline F(\alpha)$3 denotes the number of lower ideals of size $\overline F(\alpha)$4, then

$\overline F(\alpha)$5

equivalently,

$\overline F(\alpha)$6

when $\overline F(\alpha)$7 has an even number of parts (Elizalde et al., 2022).

Within the oriented-poset formalism, the circular rank polynomial is

$\overline F(\alpha)$8

and one shows in general that $\overline F(\alpha)$9 is symmetric and invariant under cyclic rotation of the parts of $x_R\succ x_L$0 (Oğuz, 2022). This cyclic invariance is specific to the loop setting and reflects the closure of the zig–zag into a cycle.

The unimodality picture is subtler. For linear fences, the interlacing results of McConville, Sagan and Smyth are deduced once symmetry for circular fences is established, and this proves unimodality for all linear $x_R\succ x_L$1 (Oğuz et al., 2021). For circular fences, the same source conjectures that unimodality holds except in some particular cases. More precisely, if the total number of nodes is odd then $x_R\succ x_L$2 is always unimodal; if the total number of nodes is $x_R\succ x_L$3, then $x_R\succ x_L$4 for all $x_R\succ x_L$5, so any dip can occur only in the middle; and the only known counter-examples are compositions of the form $x_R\succ x_L$6 or its cyclic shifts, with rank-sequence

$x_R\succ x_L$7

(Oğuz et al., 2021). The matrix-based treatment states, in parallel, that $x_R\succ x_L$8 is unimodal except for two small infinite families (Oğuz, 2022). A plausible implication is that these descriptions refer to the same obstruction pattern.

Closed forms are rare. One special case recorded for alternating compositions $x_R\succ x_L$9 expresses the circular rank polynomial using Chebyshev polynomials of the first kind, while in general no closed form is known (Oğuz et al., 2021).

4. Bijective proofs and distributive-lattice structure

The distributive-lattice viewpoint packages loop fences into the ranked lattice of lower ideals. For $R^\circ(a;q)$0, let

$R^\circ(a;q)$1

The bijective proof of full rank symmetry constructs a size-preserving bijection $R^\circ(a;q)$2 sending each ideal $R^\circ(a;q)$3 of size $R^\circ(a;q)$4 to a filter $R^\circ(a;q)$5 of size $R^\circ(a;q)$6 (Elizalde et al., 2022).

The construction has three phases. First, one encodes an ideal by the cardinalities of its intersections with alternating ascending segments $R^\circ(a;q)$7 and descending segments $R^\circ(a;q)$8: $R^\circ(a;q)$9 Filters are encoded similarly by $R^c(q;\beta)$0. In PHC1, for each $R^c(q;\beta)$1 with $R^c(q;\beta)$2 and $R^c(q;\beta)$3, one decreases $R^c(q;\beta)$4 by $R^c(q;\beta)$5 and increases $R^c(q;\beta)$6 by $R^c(q;\beta)$7. In PHC2, one studies the circular sequence $R^c(q;\beta)$8; if there is some $R^c(q;\beta)$9 with $\overline R(\alpha;q)$0, the circle is cut open at all such $\overline R(\alpha;q)$1 and the “gate-involution” $\overline R(\alpha;q)$2 is applied separately to each linear piece, while otherwise $\overline R(\alpha;q)$3 is applied to the entire circular block. The involution $\overline R(\alpha;q)$4 is itself defined by the local operations P1–P2. In PHC3, for each $\overline R(\alpha;q)$5 with the new $\overline R(\alpha;q)$6 but $\overline R(\alpha;q)$7, one replaces $\overline R(\alpha;q)$8 (Elizalde et al., 2022).

The result is a bijection preserving total size, and hence a proof that $\overline R(\alpha;q)$9. The same work also states a partial symmetry phenomenon for linear fences with an odd number of parts: the number of ideals of $F(a)$00 of size $F(a)$01 equals the number of filters of size $F(a)$02 when $F(a)$03 is below a certain value (Elizalde et al., 2022). This places loop fences within a broader program of symmetry phenomena for fence distributive lattices.

5. Rowmotion, circular tilings, and homomesy

Rowmotion on the lower-ideal lattice of a loop fence is defined in the usual way: for $F(a)$04, rowmotion $F(a)$05 sends an ideal $F(a)$06 to the lower ideal generated by the minimal elements of the complement $F(a)$07. This is a permutation of $F(a)$08 (Oğuz et al., 2021).

A combinatorial encoding of rowmotion orbits uses circular $F(a)$09-tilings: an infinite periodic $F(a)$10 tiling by colored tiles—yellow $F(a)$11, black $F(a)$12, and red $F(a)$13, with wrap-around allowed—subject to alternation and wrap-around rules. There is a bijection between $F(a)$14-orbits on $F(a)$15 and circular $F(a)$16-tilings (Oğuz et al., 2021). Two natural orbit statistics are

$F(a)$17

In the tiling model these become tile counts.

The homomesy statements for circular fences parallel those for linear fences. If $F(a)$18 are two unshared elements in the same linear segment, then $F(a)$19 is $F(a)$20-mesic on every orbit. If $F(a)$21 is an unshared node between a maximal element $F(a)$22 and a minimal element $F(a)$23, then $F(a)$24 is $F(a)$25-mesic. If $F(a)$26 is a shared maximal bead between segments $F(a)$27 and $F(a)$28 is a shared minimal bead between $F(a)$29, then $F(a)$30 is $F(a)$31-mesic, provided a small compatibility condition is met. Finally, the total

$F(a)$32

is $F(a)$33-mesic (Oğuz et al., 2021). These results place loop fences among the posets for which rowmotion admits a detailed orbit-statistical description.

6. Examples and relation to $F(a)$34-deformed Markov numbers

A basic matrix example is the ordinary fence $F(a)$35. Using the chain matrices,

$F(a)$36

and multiplication yields

$F(a)$37

Closing the loop gives

$F(a)$38

obtained as the trace of the same product matrix (Oğuz, 2022).

A second example is $F(a)$39, which has $F(a)$40 elements, obtained from the linear fence on $F(a)$41 nodes by identifying $F(a)$42. Its circular rank polynomial is

$F(a)$43

which is symmetric and unimodal, with unique peak $F(a)$44 at $F(a)$45 and $F(a)$46. In the worked rowmotion example, a sample orbit has period $F(a)$47, and the tiling model exhibits the homomesy relations explicitly (Oğuz et al., 2021).

The most prominent external connection is to $F(a)$48-deformed Markov numbers. In Leclerc–Morier-Genoud, the $F(a)$49-Markov numbers arise as

$F(a)$50

where $F(a)$51 is a Christoffel word and $F(a)$52 is built by ordinary matrix multiplication from the Cohn matrices

$F(a)$53

Each such $F(a)$54 is exactly the rank matrix of some ordinary fence poset, and therefore

$F(a)$55

where $F(a)$56 is obtained by replacing each $F(a)$57 and $F(a)$58 in the part-sequence of $F(a)$59. In particular, the $F(a)$60-Markov number attached to $F(a)$61 is the circular rank polynomial of a loop-fence (Oğuz, 2022). The same framework is also used to resolve a conjecture of Leclere and Morier-Genoud and to derive several identities between circular rank polynomials (Oğuz, 2022).