Frozen-Square Enumeration

• Frozen-square enumeration is a framework that counts configurations by imposing fixed square constraints on structures like tableaux, matrices, and lattice walks.
• It employs explicit product formulas and Fredholm determinant representations to refine counts in alternating sign matrices, standard Young tableaux, and pattern-avoiding matrices.
• The approach bridges combinatorics, statistical mechanics, and graph theory, revealing asymptotic universalities and novel links to limit shapes and random matrix phenomena.

Frozen-square enumeration is a term encompassing diverse combinatorial frameworks in which enumerative counts are sought for configurations—such as tableaux, matrices, lattice walks, permutations, partitions, or graph colorings—that include explicit constraints associated with “freezing” a square region, typically in the form of removing, fixing, or avoiding a corner, quadrant, or block. These enumerative problems have led to the discovery of product formulas, Fredholm determinant representations, structural results in reconfiguration theory, and new links to universal fluctuation phenomena.

1. Combinatorial Settings of Frozen-square Enumeration

The notion of frozen-square enumeration appears across several archetypes:

• Young Tableaux: Counting standard Young tableaux (SYT) of classical shapes (shifted staircase, rectangular) with a square region removed from the NE corner (Adin et al., 2010).
• Matrix Avoidance: Enumerating (0,1)-matrices (or permutations, polyominoes) that avoid forbidden (often square) submatrices or patterns (Ju et al., 2011, Duchi, 2019).
• Lattice Walks: Calculating the number of lattice walks that avoid entry into a forbidden quadrant (“frozen square”), leading to generating functions with both D-finite and algebraic components (Bousquet-Mélou, 2015).
• Plane Partitions and Square-ice Models: Enumerating height function configurations on square grids under local constraints analogous to plane partitions, with frozen boundaries or corners (Govindarajan et al., 2016).
• ASMs (Alternating Sign Matrices): Counting ASMs refined by the presence of an $s\times s$ block of zeroes (“frozen-corner”) and expressing such enumerations via Fredholm determinants (Colomo et al., 17 Sep 2025).
• Graph Colorings: Studying “frozen” $k$-colorings and clique partitions in $2K_2$-free graphs (the square being the cycle $C_4$ or its complement), including operations that propagate frozen blocks (Belavadi et al., 20 Sep 2024).

These settings generalize the concept of a “frozen region,” allowing enumeration under complex geometric or algebraic constraints.

2. Product and Determinant Formulas for Truncated and Frozen Shapes

In the context of tableaux and ASMs, remarkable explicit formulas govern frozen-square enumerations:

• SYT of Truncated Shapes: For shifted staircase shapes $[m]$ or rectangular shapes $(n^m)$ with a $k \times k$ square removed from the NE corner, the number of fillings admits “unexpected product formulas” involving ratios of factorials (Adin et al., 2010). For example,

$$2\, g^{p \cup X^c} = g^{[m]} \frac{(M+2|p|+1)! |p|!}{(M+|p|)! (2|p|+1)!}$$

Here, $g^{[m]}$ is Schur’s formula for the shifted staircase, $p$ encodes the shape of the removed square, and $M=\binom{m+1}{2}$.

• ASMs with Frozen Corners: Recent work provides a conjectural formula for the number $B_{n,s}$ of $n\times n$ ASMs with an $s\times s$ block of zeroes in a corner in terms of the Fredholm determinant of an explicit $s \times s$ matrix $M$ (Colomo et al., 17 Sep 2025):

$B_{n,s} = A_n \cdot \det(1 - M)$

where $A_n$ is the total number of ASMs, and the entries $M_{ij}$ are given via contour integrals or combinatorial sums depending on refined ASM counts.

These formulas unify classical results (e.g., the plain enumeration of tableaux, MacMahon’s formula for plane partitions) with new structures imposed by frozen constraints.

3. Matrix Avoidance, Block Structure, and Generating Function Techniques

In the paper of $0/1$ matrices and related configurations, frozen-square enumeration is equivalent to avoidance of forbidden $2\times 2$ submatrices. Central results include (Ju et al., 2011):

• Block Forms and Set Partitions: For certain avoidance conditions (e.g., avoiding the matrix $\Gamma$ or $C$), matrices can be transformed by row/column permutations into block-diagonal forms; enumeration then reduces to sums over pairs of set partitions and permutations.
• Rook Placement Correspondence: In {T, L}-avoiding matrices, the count reduces to enumeration of nonattacking rook placements, leading to

$$\varphi(k, n; \{T, L\}) = 2 \sum_{m=0}^{\min(k, n)} \binom{k}{m} \binom{n}{m} m!$$

• Exponential Generating Functions (EGF): For each avoidance class, closed EGF formulas encapsulate the enumerative complexity, for example,

$$\Phi(x, y; I) = \frac{\exp(x + y)}{1 - (\exp(x) - 1)(\exp(y) - 1)}$$

These approaches generalize to permutation patterns and convex polyominoes via encoding schemes, leading to unified rational/algebraic generating functions (Duchi, 2019).

4. Lattice Path Models and Forbidden Quadrants

In lattice walk enumeration, the notion of freezing a square region translates to avoidance of entry into a non-convex quadrant (Bousquet-Mélou, 2015). Key findings include:

• Generating Function Structure: For walks confined to a cone $C$, the generating function decomposes as

$C(x, y) = \frac{1}{3} Q(x, y) + P(x, y)$

where $Q(x, y)$ is the standard D-finite function for quadrant walks, and $P(x, y)$ is algebraic (often degree 72).

• Closed-form Counts: Endpoint enumeration admits hypergeometric closed-form expressions in certain cases.
• Reflection Principle: Many frozen-square enumeration formulas are linked to classic quadrant models (e.g., Gessel’s walks) by reflection identities,

$C_{i, j} = Q_{i, j} + C_{-i-2, j} + C_{i, -j-2}$

The frozen-square condition alters both leading asymptotics and generating function properties, producing new universality classes.

5. Statistical Mechanical Models: Plane Partitions, Square Ice, and Limit Shapes

Frozen-square enumeration is deeply entwined with statistical mechanics and random matrix theory through models such as plane partitions and square ice (Govindarajan et al., 2016, Ayyer et al., 2012, Colomo et al., 17 Sep 2025):

• Enumerative Algorithms: The Bratley–McKay algorithm and transition matrix Monte Carlo methods allow exact and asymptotic enumeration of square-ice analogues of plane partitions.
• Asymptotic Universalities: For these models, the asymptotic behavior of the enumerated counts is well-described by stretched-exponential laws:

$$\log a_\ell(n) \sim c_0 n^{2/3} + c_1 \log n + c_2 + c_3 n^{1/3}$$

with $c_0$ independent of the parameter $\ell$.

• Limit Shape and Tracy–Widom Fluctuations: In the frozen-corner refinement of ASMs, the Fredholm determinant structure leads (under proper scaling) to the Tracy–Widom distribution for edge fluctuations, linking combinatorial enumeration to universal behaviors in random matrices.

6. Graph Coloring, Clique Partitioning, and Frozen Configurations

Frozen-square enumeration extends to graph theory via the paper of frozen colorings and clique partitions in $2K_2$-free graphs (Belavadi et al., 20 Sep 2024):

• Frozen Coloring Characterization: A $k$-coloring is frozen if each vertex is adjacent to every other color in its neighborhood; equivalently, in clique partition language, each vertex must have a non-neighbor in every other clique.
• Classes and Operations: Infinite families of such graphs (e.g., ME$_q$) are constructed with recursive operations that increase chromatic number while preserving the frozen property and forbidden square subgraph (i.e., $C_4$ or $2K_2$-freeness).
• Enumeration and Complexity: Results show for every $k \geq 4$ there exists a $k$-chromatic $2K_2$-free graph with a frozen $(k+1)$-coloring, narrowing the gap between chromatic number and frozen coloring number.

7. Broader Implications, Applications, and Open Problems

Frozen-square enumeration yields both exact and asymptotic results, product and determinant formulas, and bridges between combinatorics, statistical mechanics, and algebraic structures.

• Applications: These include counting configurations in coding theory (matrix avoidance), statistical physics (limit shapes and phase boundaries), and algorithmic random generation (grid permutations, polyominoes).
• Structural Insights: Product formulas uncover hidden factorization phenomena, while determinant formulas and reflection principles reveal connections between boundary constraints and universality classes.
• Open Directions: Active research surrounds combinatorial interpretations of algebraic/reflection identities, classification of step sets in non-convex walk regions, further exploration of Fredholm determinant connections, and graph-theoretic dichotomies for recolourability and frozen configurations.

Frozen-square enumeration thus constitutes a fertile paradigm for combinatorial analysis, linking explicit enumeration to universal behaviors and algebraic structures, and revealing deep mathematical phenomena underlying simple frozen constraints.