--- title: Rectangular Paths Conjecture url: https://www.emergentmind.com/topics/rectangular-paths-conjecture type: topic --- # Rectangular Paths Conjecture 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 [1809.03070], [2206.00131], [2205.08025], [2006.14424], [2508.20935]. ## 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 \(R_1,\dots,R_4\) lie on \(\gamma\) with cyclic order matching the counterclockwise order on \(\gamma\). The conjectural statement is that there exists a continuous path \[ R_t,\quad t\in[0,1], \] of labeled rectangles gracing \(\gamma\) such that \(R_0\) is degenerate of aspect ratio \(X=0\), \(R_1\) is degenerate of aspect ratio \(Y=0\), each intermediate rectangle has both side-lengths \(X(R_t),Y(R_t)>0\), and the cyclic ordering of its vertices agrees with that of \(\gamma\). The associated expectation is stronger: the union of all vertex-sets \(\{R_t\}\) should cover all but finitely many points of \(\gamma\) [1809.03070]. The configuration space is expressed as \(\Gamma(\gamma)\), the space of gracing rectangles. A **sweepout** is a continuous piecewise-smooth path \(\alpha\subset \Gamma(\gamma)\). Such an arc is **hyperbolic** if it connects a rectangle of aspect ratio \(X=0\) to one of ratio \(Y=0\); otherwise it is a **null arc**, meaning that both ends lie on the same axis in shape-space. If \(R_t\) parametrizes \(\alpha\), its **shape curve** is \[ Z(\alpha)=\{(X(R_t),Y(R_t)):t\in[0,1]\}\subset \mathbb R^2. \] For polygons, the relevant combinatorial datum is the number \(\Delta_+(P)\) of **positively oriented diameters**, also called positively oriented extremal chords. The generic polygonal setting uses a fat open subset \(\Pi_N^\ast\subset \Pi_N\) of embedded \(N\)-gons for which \(\Gamma(P)\) is a \(1\)-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 \((X,Y)\); the total coincidence count is \[ M(P)=\sum_{(X,Y)}\mu(X,Y), \] where \(\mu(X,Y)=n-1\) if exactly \(n>1\) distinct rectangles realize that shape and \(\mu=0\) otherwise [1809.03070]. ## 3. Integral invariant, shape-space, and coincidence growth The central analytic tool is an integral formula attached to a piecewise-smooth path \(R_t\) of gracing rectangles. Writing \[ X_t=\mathrm{length}(R_{t,1}R_{t,2}),\qquad Y_t=\mathrm{length}(R_{t,2}R_{t,3}), \] one defines four caps \(R_j^\ast\) bounded by an edge \(R_jR_{j+1}\) together with the corresponding arc of the loop, and lets \(A_j(t)\) be the signed area of \(R_j^\ast\). The alternating cap-area invariant is \[ A(t)=[A_1(t)+A_3(t)]-[A_2(t)+A_4(t)]. \] A local calculation gives \[ \frac{dA}{dt}=Y_t\frac{dX}{dt}-X_t\frac{dY}{dt}. \] Equivalently, with \(\omega=-X\,dY+Y\,dX\), \[ A(1)-A(0)=\int_{t=0}^1 [Y\,dX-X\,dY]=\int_{Z(\alpha)}\omega. \] If the shape curve is closed by straight segments from its endpoints to the origin, the resulting augmented loop encloses a signed region \(\Omega\subset \mathbb R^2\) satisfying \[ \oint_Z\omega=\oint_{Z'}\omega=-2\cdot \mathrm{Area}(\Omega). \] Theorem 1.5 then identifies the geometric meaning of this invariant: for a closed loop or a null arc, \(\int\omega=0\) and the bounded region has area \(0\); for a hyperbolic arc, \(\int\omega=\pm2\cdot \mathrm{Area}(J)\), and the augmented shape-loop encloses exactly \(\mathrm{Area}(J)\) [1809.03070]. This integral formula drives the counting theorem for polygons. Schwartz’s Theorem 1.8 states that for every \(P\in \Pi_N^\ast\), \[ M(P)\ge 2\cdot(\Delta_+(P)-2). \] 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 \(\alpha_1,\dots,\alpha_d\) are the hyperbolic arcs with embedded shape-loops, each loop encloses exactly \(\mathrm{Area}(P)\), so any two of them must intersect in the positive quadrant; an inductive argument then yields at least \(d-1\) new coincidences. Altogether this implies linear growth of \(M(P)\) in the number of positively oriented diameters [1809.03070]. The convex polygon example makes the mechanism explicit. For a strictly convex \(N\)-gon, rotating-calipers gives \(\Delta_+(P)=N\), hence \[ M(P)\ge 2(N-2). \] In the unlabeled count this becomes \(M(P)\ge N-2\). Each diameter determines a hyperbolic arc, and the associated shape curve is a branch of the hyperbola \[ X\cdot Y=\mathrm{Area}(P). \] There are no null arcs in the convex case. The integral formula then becomes transparent along each hyperbola branch, where \(-X\,dY+Y\,dX\) integrates to \(\pm2\cdot \mathrm{Area}(P)\). The paper presents this as strong evidence for the full sweepout conjecture in the general Jordan setting [1809.03070]. ## 4. Symmetric functions and the rectangular lattice-path conjecture In algebraic combinatorics, the Rectangular Paths Conjecture is a \(q,t\)-enumerative identity for labeled lattice paths in an \(m\times n\) rectangle. A **rectangular path** is a lattice path from \((0,0)\) to \((m,n)\) 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 \(my=nx\). The sets are denoted \(RP(m,n)\) and \(RD(m,n)\). If the \(N\)-steps are ordered \(1,\dots,n\), and \(a_i=x_i-\frac{m}{n}y_i\) is the signed horizontal distance of the start of the \(i\)-th \(N\)-step from the diagonal, then the **shift** is \[ s(\pi)=-\min_{1\le i\le n} a_i, \] the **base diagonal** is \(y=\frac{n}{m}(x-s(\pi))\), and the **area** is \[ \mathrm{area}(\pi)=\sum_{i=1}^n \lfloor a_i+s(\pi)\rfloor. \] The attack relation is defined by \[ (a_i,i)<_{\mathrm{lex}}(a_j,j)<_{\mathrm{lex}}(a_i+\tfrac{m}{n},i), \] and for a labeling \(w=(w_1,\dots,w_n)\) one sets \[ \mathrm{tdinv}(\pi,w)=\#\{1\le i1\), the general conjecture remains open, and extensive computer checks verify it up to \(m+n\le 13\) [2206.00131]. ## 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 \(1\), and in other columns slope \(n/m\). Vertical distances are recomputed from this broken diagonal, the area is taken by summing \(\lfloor v_i+s\rfloor\) over non-decorated East steps, and the dinv statistic is modified by a new correction term \(\mathrm{cdinv}(\pi,D)\) that adds and subtracts intersections between vertical steps and decorated falls. The resulting theorem for Dyck objects is unconditional: \[ s_{(m-1)^k}^{\perp}\,e_{m,n+km} = \sum_{\pi\in LRD(m+k,n+k)_{\ast k}} q^{\mathrm{dinv}(\pi)}\,t^{\mathrm{area}(\pi)}\,x^\pi. \] The corresponding rectangular statement is conditional on the Rectangular Paths Conjecture: \[ s_{(m-1)^k}^{\perp}\Bigl(\frac{[m]_q}{[d]_q}\,p_{m,n+km}\Bigr) = \sum_{\pi\in LRP(m+k,n+k)_{\ast k}} q^{\mathrm{dinv}(\pi)}\,t^{\mathrm{area}(\pi)}\,x^\pi. \] In particular, this conditional statement holds unconditionally whenever \(d=1\), because the coprime rectangular conjecture is already proved. The proof strategy for the unconditional Dyck theorem uses a skew-operator expansion for \(s_{(m-1)^k}\), a combinatorial bijection between \(h_{\tilde\alpha}^{\perp}\)-restrictions of \(LRP(m,n+km)\) 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 [2508.20935]. ## 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 \[ A:y=m_Ax+b_A,\quad B:y=m_Bx+b_B,\quad C:y=m_Cx,\quad D:y=m_Dx, \] over an arbitrary field \(k\). 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 \[ \pi:\mathbb P^1_k\to \mathbf C,\qquad \phi:\mathbb P^1_k\to \mathbf C, \] the first parameterized by slope and the second by aspect ratio. In the non-degenerate case, meaning that the two diagonals \(E\) and \(F\) of the complete quadrilateral are not orthogonal, the two images coincide and form an irreducible plane conic of genus \(0\); in the degenerate case \(E\perp F\), 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-\(2\) formulas for both parametrizations, and constructs the homography \(\Psi:\mathbb P^1_k\to\mathbb P^1_k\) satisfying \(\pi=\phi\circ \Psi\) and \(\phi=\pi\circ \Psi^{-1}\). In this special setting, what one might call the Rectangular Paths Conjecture is therefore a theorem [2006.14424]. A different usage appears in the theory of Hamiltonian-path reconfiguration in rectangular grid graphs. There, a **simple** \(s,t\) path is an \(s,t\) Hamiltonian path in an \(m\times n\) grid graph such that every internal subpath with boundary endpoints uses the minimum possible number of bends: \(0\) for opposite sides, \(1\) for adjacent sides, and \(2\) for the same side. The unique local move is the \(2\times 2\) **square-switch**, defined on a switchable square straddling a zipline, main track, and side track. The resulting reconfiguration graph \(\mathcal G\), whose vertices are all simple \(s,t\) Hamiltonian paths and whose edges correspond to one square-switch, is proved to be connected, with \[ \mathrm{diam}(\mathcal G)\le 5|\mathbb G|/4, \] and this bound is asymptotically tight. Each square-switch is done in \(O(1)\) time, each zip runs in \(O(\max\{m,n\})\) time, and the overall reconfiguration from one simple path to another is carried out in \(O(|\mathbb G|)\) 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 [2205.08025]. ## 7. Conceptual relations and open directions Across its different meanings, the phrase organizes several recurrent structures: \(1\)-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 \(\int (Y\,dX-X\,dY)\), which forces hyperbolic arcs to enclose \(\mathrm{Area}(J)\) in shape-space and yields lower bounds on coincidences [1809.03070]. In the lattice-path setting, the decisive invariant is the symmetric-function side \(\frac{[m]_q}{[d]_q}p_{m,n}\), together with the interplay of area, dinv, sweep maps, and Macdonald-operator formalisms [2206.00131]. In the fall-decorated extension, the same framework persists after introducing a broken diagonal and a corrected dinv term [2508.20935]. The principal open case remains the non-coprime algebraic-combinatorial conjecture for \(d=\gcd(m,n)>1\). By contrast, the four-line version is settled completely, and the grid-graph reconfiguration version is also settled with explicit diameter and algorithmic bounds [2006.14424], [2205.08025]. 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 \[ M(P)\ge 2(\Delta_+(P)-2), \] 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 [1809.03070]. 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. Source: https://www.emergentmind.com/topics/rectangular-paths-conjecture