Rectangular Paths Conjecture

• Rectangular Paths Conjecture is an umbrella term covering distinct formulations that link inscribed rectangles in Jordan loops, lattice paths, and grid graph reconfigurations.
• The conjecture employs integral invariants, symmetric function identities, and combinatorial sweep maps to establish bounds, count coincidences, and parameterize rectangular paths.
• Recent advances have proved key cases for coprime settings and grid graphs, while general non-coprime and sweepout conjectures remain open challenges.

Searching arXiv for papers explicitly using “Rectangular Paths Conjecture” and closely related formulations. The expression Rectangular Paths Conjecture does not designate a single universally standardized statement. In current arXiv usage it refers to several mathematically distinct formulations: a sweepout conjecture for inscribed rectangles in piecewise-smooth Jordan loops, a symmetric-function identity for labeled rectangular lattice paths, and, in expository summaries, a connectivity statement for reconfiguration of simple Hamiltonian paths in rectangular grid graphs; a closely related four-line incidence problem has been proved completely over an arbitrary field (Schwartz, 2018, Iraci et al., 2022, Nishat et al., 2022, Olberding et al., 2020, Iraci et al., 28 Aug 2025).

1. Terminological scope

The phrase is best understood as an overloaded label whose precise meaning is determined by context. This suggests that the term functions more as a family resemblance across “path” problems involving rectangles than as a single conjecture with one canonical formulation.

Setting	Core statement	Status
Inscribed rectangles in Jordan loops	Existence of a continuous path of gracing rectangles from aspect ratio $X=0$ to $Y=0$	Conjectural in general; supported by integral and counting results
Labeled rectangular lattice paths	$$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$	Proved when $\gcd(m,n)=1$; open for $d>1$
Simple $s,t$ paths in rectangular grids	All simple $s,t$ Hamiltonian paths are connected by $2\times2$ square-switches	Proved with diameter at most $5|\mathbb G|/4$
Rectangles inscribed in four lines	All such rectangles lie in two rational families, generically the same conic	Proved over an arbitrary field

2. Sweepouts of inscribed rectangles

In the inscribed-rectangle setting, the conjecture concerns a piecewise-smooth, positively oriented Jordan loop $\gamma$ in the plane. A gracing or inscribed rectangle is a possibly degenerate rectangle whose four vertices $Y=0$0 lie on $Y=0$1 with cyclic order matching the counterclockwise order on $Y=0$2. The conjectural statement is that there exists a continuous path

$Y=0$3

of labeled rectangles gracing $Y=0$4 such that $Y=0$5 is degenerate of aspect ratio $Y=0$6, $Y=0$7 is degenerate of aspect ratio $Y=0$8, each intermediate rectangle has both side-lengths $Y=0$9, and the cyclic ordering of its vertices agrees with that of $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$0. The associated expectation is stronger: the union of all vertex-sets $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$1 should cover all but finitely many points of $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$2 (Schwartz, 2018).

The configuration space is expressed as $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$3, the space of gracing rectangles. A sweepout is a continuous piecewise-smooth path $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$4. Such an arc is hyperbolic if it connects a rectangle of aspect ratio $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$5 to one of ratio $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$6; otherwise it is a null arc, meaning that both ends lie on the same axis in shape-space. If $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$7 parametrizes $$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$8, its shape curve is

$$\frac{[m]_q}{[d]_q}p_{m,n}=\sum q^{\mathrm{dinv}}t^{\mathrm{area}}x^\pi$$9

For polygons, the relevant combinatorial datum is the number $\gcd(m,n)=1$0 of positively oriented diameters, also called positively oriented extremal chords. The generic polygonal setting uses a fat open subset $\gcd(m,n)=1$1 of embedded $\gcd(m,n)=1$2-gons for which $\gcd(m,n)=1$3 is a $\gcd(m,n)=1$4-manifold of piecewise-smooth arcs, each arc joins two positive diameters, and each diameter is an endpoint of exactly four arcs under cyclic relabeling. Within this setting, the paper studies rectangle coincidences, namely really-distinct labeled rectangles that are isometric because they share the same pair $\gcd(m,n)=1$5; the total coincidence count is

$\gcd(m,n)=1$6

where $\gcd(m,n)=1$7 if exactly $\gcd(m,n)=1$8 distinct rectangles realize that shape and $\gcd(m,n)=1$9 otherwise (Schwartz, 2018).

3. Integral invariant, shape-space, and coincidence growth

The central analytic tool is an integral formula attached to a piecewise-smooth path $d>1$0 of gracing rectangles. Writing

$d>1$1

one defines four caps $d>1$2 bounded by an edge $d>1$3 together with the corresponding arc of the loop, and lets $d>1$4 be the signed area of $d>1$5. The alternating cap-area invariant is

$d>1$6

A local calculation gives

$d>1$7

Equivalently, with $d>1$8,

$d>1$9

If the shape curve is closed by straight segments from its endpoints to the origin, the resulting augmented loop encloses a signed region $s,t$0 satisfying

$s,t$1

Theorem 1.5 then identifies the geometric meaning of this invariant: for a closed loop or a null arc, $s,t$2 and the bounded region has area $s,t$3; for a hyperbolic arc, $s,t$4, and the augmented shape-loop encloses exactly $s,t$5 (Schwartz, 2018).

This integral formula drives the counting theorem for polygons. Schwartz’s Theorem 1.8 states that for every $s,t$6,

$s,t$7

The proof distinguishes null arcs, non-embedded hyperbolic arcs, and embedded hyperbolic arcs. A null arc has zero enclosed area in shape-space, forcing a self-intersection of its shape curve and hence at least one rectangle coincidence. A non-embedded hyperbolic arc also yields a self-intersection. If $s,t$8 are the hyperbolic arcs with embedded shape-loops, each loop encloses exactly $s,t$9, so any two of them must intersect in the positive quadrant; an inductive argument then yields at least $s,t$0 new coincidences. Altogether this implies linear growth of $s,t$1 in the number of positively oriented diameters (Schwartz, 2018).

The convex polygon example makes the mechanism explicit. For a strictly convex $s,t$2-gon, rotating-calipers gives $s,t$3, hence

$s,t$4

In the unlabeled count this becomes $s,t$5. Each diameter determines a hyperbolic arc, and the associated shape curve is a branch of the hyperbola

$s,t$6

There are no null arcs in the convex case. The integral formula then becomes transparent along each hyperbola branch, where $s,t$7 integrates to $s,t$8. The paper presents this as strong evidence for the full sweepout conjecture in the general Jordan setting (Schwartz, 2018).

4. Symmetric functions and the rectangular lattice-path conjecture

In algebraic combinatorics, the Rectangular Paths Conjecture is a $s,t$9-enumerative identity for labeled lattice paths in an $2\times2$0 rectangle. A rectangular path is a lattice path from $2\times2$1 to $2\times2$2 using North and East steps and ending in an East step; a rectangular Dyck path is such a path that never goes strictly below the main diagonal $2\times2$3. The sets are denoted $2\times2$4 and $2\times2$5. If the $2\times2$6-steps are ordered $2\times2$7, and $2\times2$8 is the signed horizontal distance of the start of the $2\times2$9-th $5|\mathbb G|/4$0-step from the diagonal, then the shift is

$5|\mathbb G|/4$1

the base diagonal is $5|\mathbb G|/4$2, and the area is

$5|\mathbb G|/4$3

The attack relation is defined by

$5|\mathbb G|/4$4

and for a labeling $5|\mathbb G|/4$5 one sets

$5|\mathbb G|/4$6

The dinv correction is

$5|\mathbb G|/4$7

and the full statistic is

$5|\mathbb G|/4$8

On the symmetric-function side, with $5|\mathbb G|/4$9, one defines $\gamma$0 and $\gamma$1 through the Schiffmann–Vasserot–Bergeron–Garsia operators $\gamma$2 and the linear maps $\gamma$3 for coprime $\gamma$4 and $\gamma$5. The conjecture is

$\gamma$6

In the coprime case $\gamma$7, this becomes

$\gamma$8

Iraci, Pagaria, Paolini, and Vanden Wyngaerd prove this coprime case by combining Mellit’s Rectangular Shuffle Theorem

$\gamma$9

with a sweep process extending from Dyck paths to all rectangular paths, together with a cyclic decomposition of $Y=0$00 into cycles of length $Y=0$01. Under the cyclic shift $Y=0$02, the sweep weight transforms by

$Y=0$03

where the exponents run through $Y=0$04, yielding the factor $Y=0$05. The same work situates the conjecture relative to the shuffle theorem, the square paths theorem, and rectangular analogues of the rise-Delta and Delta square conjectures at $Y=0$06. For $Y=0$07, the general conjecture remains open, and extensive computer checks verify it up to $Y=0$08 (Iraci et al., 2022).

5. Fall-decorated extensions and conditional consequences

A later development introduces fall-decorated rectangular paths and uses the rectangular-paths identity as a hypothesis in the non-coprime case. Here a fall is an East step followed immediately by another East step, and some falls are decorated. The geometry of the path is then measured against a broken diagonal: in each column with a decorated fall one uses slope $Y=0$09, and in other columns slope $Y=0$10. Vertical distances are recomputed from this broken diagonal, the area is taken by summing $Y=0$11 over non-decorated East steps, and the dinv statistic is modified by a new correction term $Y=0$12 that adds and subtracts intersections between vertical steps and decorated falls. The resulting theorem for Dyck objects is unconditional: $Y=0$13 The corresponding rectangular statement is conditional on the Rectangular Paths Conjecture: $Y=0$14 In particular, this conditional statement holds unconditionally whenever $Y=0$15, because the coprime rectangular conjecture is already proved. The proof strategy for the unconditional Dyck theorem uses a skew-operator expansion for $Y=0$16, a combinatorial bijection between $Y=0$17-restrictions of $Y=0$18 and fall-decorated paths, a sign-reversing involution on fall-labels, and a careful tracking of the dinv correction. In this literature the rectangular conjecture is explicitly presented as the non-coprime input needed to extend the fall-decorated theory from Dyck paths to all rectangular paths (Iraci et al., 28 Aug 2025).

6. Specialized formulations: four lines and rectangular grid graphs

A fully solved geometric special case concerns rectangles with one vertex on each of four lines

$Y=0$19

over an arbitrary field $Y=0$20. Olberding and Walker show that the overall locus of such rectangles is a plane conic in the five-dimensional projective space of parallelograms, and they construct two rational maps

$Y=0$21

the first parameterized by slope and the second by aspect ratio. In the non-degenerate case, meaning that the two diagonals $Y=0$22 and $Y=0$23 of the complete quadrilateral are not orthogonal, the two images coincide and form an irreducible plane conic of genus $Y=0$24; in the degenerate case $Y=0$25, the conic splits into two lines, one the slope-path and the other the aspect-path. The paper also characterizes rectangles at infinity, gives explicit degree-$Y=0$26 formulas for both parametrizations, and constructs the homography $Y=0$27 satisfying $Y=0$28 and $Y=0$29. In this special setting, what one might call the Rectangular Paths Conjecture is therefore a theorem (Olberding et al., 2020).

A different usage appears in the theory of Hamiltonian-path reconfiguration in rectangular grid graphs. There, a simple $Y=0$30 path is an $Y=0$31 Hamiltonian path in an $Y=0$32 grid graph such that every internal subpath with boundary endpoints uses the minimum possible number of bends: $Y=0$33 for opposite sides, $Y=0$34 for adjacent sides, and $Y=0$35 for the same side. The unique local move is the $Y=0$36 square-switch, defined on a switchable square straddling a zipline, main track, and side track. The resulting reconfiguration graph $Y=0$37, whose vertices are all simple $Y=0$38 Hamiltonian paths and whose edges correspond to one square-switch, is proved to be connected, with

$Y=0$39

and this bound is asymptotically tight. Each square-switch is done in $Y=0$40 time, each zip runs in $Y=0$41 time, and the overall reconfiguration from one simple path to another is carried out in $Y=0$42 time. In expository accounts, this result is described as settling a Rectangular Paths Conjecture for grid graphs, but its content is reconfiguration-theoretic rather than geometric or symmetric-function-theoretic (Nishat et al., 2022).

7. Conceptual relations and open directions

Across its different meanings, the phrase organizes several recurrent structures: $Y=0$43-parameter families of rectangles or rectangular objects, auxiliary shape spaces or generating functions, and invariants that constrain possible deformations. In the inscribed-rectangle problem, the decisive invariant is the cap-area integral $Y=0$44, which forces hyperbolic arcs to enclose $Y=0$45 in shape-space and yields lower bounds on coincidences (Schwartz, 2018). In the lattice-path setting, the decisive invariant is the symmetric-function side $Y=0$46, together with the interplay of area, dinv, sweep maps, and Macdonald-operator formalisms (Iraci et al., 2022). In the fall-decorated extension, the same framework persists after introducing a broken diagonal and a corrected dinv term (Iraci et al., 28 Aug 2025).

The principal open case remains the non-coprime algebraic-combinatorial conjecture for $Y=0$47. By contrast, the four-line version is settled completely, and the grid-graph reconfiguration version is also settled with explicit diameter and algorithmic bounds (Olberding et al., 2020, Nishat et al., 2022). In the inscribed-rectangle literature, the sweepout conjecture for general piecewise-smooth Jordan loops is not proved, but the integral formula, the linear-growth theorem

$Y=0$48

the existence of connected gracing sets meeting every but at most four boundary points in earlier work cited there, and the convex-polygon model together provide strong evidence for the full conjecture (Schwartz, 2018).

A common misconception is that all occurrences of the term refer to the same open problem. The literature instead supports a more precise view: the phrase names several distinct conjectural or expository programs, each centered on “rectangular paths,” but each embedded in a different ambient theory—inscribed-rectangle topology, symmetric-function combinatorics, grid-graph reconfiguration, or algebraic geometry over fields.