Petrie Matrices in Combinatorics and Topology

• Petrie matrices are structured {0,1}-matrices defined by the consecutive-ones property, encoding key combinatorial and geometric constraints.
• They determine Schur expansion coefficients in symmetric functions with determinants necessarily being -1, 0, or 1, linking to unique ribbon tilings and acyclic configurations.
• Their applications extend from combinatorial graph orientations and regular map symmetries to the classification of GKM manifolds in algebraic topology.

Petrie matrices are highly structured {0,1}-matrices central to combinatorial representation theory, map symmetry, algebraic topology of GKM manifolds, and the theory of symmetric functions. Their key feature is the “consecutive-ones” property in each row, which encodes geometric, algebraic, or combinatorial constraints depending on context. Determinants of Petrie matrices—often referred to as Petrie numbers—govern transition coefficients in Schur expansions of symmetric functions and signal symmetry invariants in polyhedral or toric topology.

1. Formal Definition and Matrix Structure

A Petrie matrix is a square (or rectangular) {0,1}-matrix whose rows are “blocks” of consecutive ones. For vectors indexed from 0 to $n-1$, a typical row is

$$v[i, j] := (0, \ldots, 0, 1, 1, \ldots, 1, 0, \ldots, 0),$$

with ones in positions $i+1$ through $j$, $0 \leq i \leq j < n$. More generally, the $k$-Petrie matrix $P_k(\lambda, \mu)$ for partitions $\lambda, \mu$ of length $n$ is defined by

$$(P_k(\lambda,\mu))_{i,j} = \left[ 0 \leq \lambda_i - \mu_j - i + j < k \right],$$

where $[\;\cdot\;]$ is the Iverson bracket (1 if condition is true, 0 otherwise) (Wu et al., 21 Sep 2025, Grinberg, 2020). This formulation originates from the determination of Schur expansion coefficients for Petrie symmetric functions, a class interpolating elementary and homogeneous symmetric functions.

The crucial property, proved by Gordon and Wilkinson, is that the determinant of a square Petrie matrix is always $0$, $1$, or $-1$. Nonzero determinants correspond combinatorially to acyclic configurations in an associated graph (“Petrie graph”), and when interpreted via ribbon-tiling they indicate unique tiling (i.e., “good orientation”) (Wu et al., 21 Sep 2025).

2. Petrie Matrices in Algebraic Combinatorics: Symmetric Functions

Petrie matrices arise naturally in Schur function expansions for the family of $k$-Petrie symmetric functions $G(k,m)$, defined via

$$\sum_{m \geq 0} G(k,m)\,z^m = \prod_{i \geq 1} (1 + x_i z + x_i^2 z^2 + \ldots + x_i^{k-1} z^{k-1}),$$

i.e., each variable’s exponent in every monomial is bounded by $k-1$ (Grinberg, 2020, Cheng et al., 2022).

The Schur expansion coefficients of $G(k,m)$ are

$$G(k,m) = \sum_{\lambda \vdash m} \operatorname{pet}_k(\lambda, \varnothing)\, s_\lambda,$$

where $$\operatorname{pet}_k(\lambda, \varnothing) = \det P_k(\lambda, \varnothing)$$ is the Petrie number, always in $\{0,1,-1\}$ (Grinberg, 2020).

The Petrie Pieri Rule for Schur functions and Petrie symmetric functions is

$$G(k,m)\cdot s_{\mu} = \sum_{\lambda \vdash m + |\mu|} \operatorname{pet}_k(\lambda, \mu)\, s_\lambda,$$

where $\operatorname{pet}_k(\lambda, \mu)$ is the determinant of a $k$-Petrie matrix formed from $\lambda, \mu$ (Wu et al., 21 Sep 2025, Jin et al., 2 Jun 2024).

For products with power sum symmetric functions $p_n$, the expansion retains signed multiplicity-freedom (i.e. coefficients in $\{-1,0,1\}$) under constraints on $k$ and $n$. For $k \geq 3$, $k$ divides $n$, and $m \geq n$, one has

$G(k,m) \cdot p_n = \sum_{*} \pm s_\lambda,$

with explicit combinatorial sign computation from rim hooks and abacus configurations (Cheng et al., 2022).

3. Combinatorial and Graph-Theoretic Interpretation

Determinants of Petrie matrices are tightly connected to combinatorial models:

• Ribbon Tiling: A determinant is nonzero iff the associated skew shape (diagram $\lambda/\mu$) admits a unique proper $k$-ribbon tiling; its sign is the parity of the sum of ribbon heights (Jin et al., 2 Jun 2024, Wu et al., 21 Sep 2025).
  • For connected diagrams where each row contains fewer than $k$ cells, and exists exactly one proper tiling, the coefficient in the Schur expansion is

  $$\operatorname{pet}_k(\lambda, \mu) = \prod_{i=1}^{m} (-1)^{r(\Theta_i)},$$

  with $r(\Theta_i)$ the number of rows crossed by the $i$-th ribbon.

• Petrie Graphs and Good Orientations: A Petrie matrix gives rise to a directed graph whose vertices correspond to rows; edges encode the consecutive ones. A “good orientation” is one where every vertex except the last has exactly one outgoing edge. Key results show that determinants can be written as signed sums over such orientations (Wu et al., 21 Sep 2025). If multiple good orientations exist, signs cancel and the determinant vanishes.
• Abacus and Rim Hook Moves: Abacus models tie the sign in rim hook removal to permutations of runner towers; only certain abacus configurations yield nonzero determinants, corresponding to partitions with $k$-core of size one or zero (Cheng et al., 2022).

Generating function results classify how statistics—such as number of reversed cycles and permutation inversions—distribute across orientation classes, explaining when the coefficient is zero or $\pm1$ (Wu et al., 21 Sep 2025).

4. Geometric Interpretations: Maps, Polyhedra, and GKM Manifolds

Petrie matrices also encode geometric and combinatorial data in:

• Regular Maps and Polyhedral Symmetries: The structure of Petrie polygons in regular maps is mirrored algebraically by Petrie matrices, with duality and Petrie-duality operations corresponding to matrix permutations. In constructing skeletal polyhedra with icosahedral symmetry, as for Gordan’s regular map and its six Petrie relatives, such matrices encode combinatorial cycles along Petrie polygons; their transformations guide duality classes and geometric realisations (Cutler et al., 2012).
• Maps on Linear Fractional Groups: Automorphism conditions for self-Petrie-duality, self-duality, and Möbius regularity in regular maps on groups like ${\rm PSL}(2,q)$ or ${\rm PGL}(2,q)$ have precise matrix interpretations relating group-theoretic involutions to matrix symmetries; verifying invariance under duality translates to invariance in the associated Petrie matrix (Erskine et al., 2018).
• GKM Manifolds and Equivariant Bundles: In torus-equivariant topology, the edge-labeling data of GKM graphs is organized into a $2\times 2$ “Petrie matrix,” used to determine characteristic classes and clutching numbers in equivariant $S^4$-bundles over $S^4$. The determinant and invariants derived from the Petrie matrix distinguish homeomorphism and diffeomorphism types within families of eight-manifolds (Goertsches et al., 3 Sep 2025). For instance, the first Pontryagin class is given by formulas such as

  $p_1(M) = -2\,(a-c)(b-d)\,\pi^*(\iota)$

  and conditions like $\det(P) \neq 0$ classify symmetries and the possible extension to higher torus actions.

5. Algebraic and Enumerative Properties

A profound result (Grinberg, 2020, Cheng et al., 2022) links Petrie matrices to algebraic independence and generating systems in the algebra of symmetric functions:

• For fixed $k$ with $1-k$ invertible in the underlying ring, the set $\{G(k,1), G(k,2), G(k,3), \ldots\}$ is algebraically independent and generates the entire symmetric function algebra.
• Any expansion coefficient for products $G(k,m) \cdot s_\mu$ or related plethysms is given by the determinant of a $k$-Petrie matrix with entries computed from $\lambda, \mu$.
• The structure constants in Pieri-type rules for Petrie symmetric functions are always in $\{0,1,-1\}$: the determinantal nature linked to the consecutive-ones property and the graph-orientation combinatorics.

Algebraic Feature	Petrie Matrix Role	Arising In
Schur Expansion Coefficient	Determinant $\operatorname{pet}_k(\lambda, \mu)$	Symmetric functions
Bundle Characteristic Class	Invariant from $2\times 2$ matrix $(a,b;c,d)$ of weights	Equivariant topology
Map Duality Symmetry	Matrix permutation, invariance under automorphisms	Regular map theory

6. Generalizations and Future Developments

Recent work generalizes Petrie matrices and their interpretation:

• Extension to arbitrary skew shapes via combinatorial graph orientations and ribbon tilings, enabling transparent proofs of plethystic Pieri rules and Murnaghan–Nakayama-type identities (Wu et al., 21 Sep 2025).
• Bivariate and multivariate generating functions tracking orientation statistics and inversion numbers, elucidating the appearance of multiplicities and sign cancellations in expansions.
• Connections to λ-ring theory and representation-theoretic specializations, particularly via plethysm with roots of unity (Jin et al., 2 Jun 2024).

The tight integration of Petrie matrices with ribbon tilings, abacus combinatorics, and symmetric function theory opens additional avenues for enumerative combinatorics, representation theory, and even topological classification of GKM manifolds and toric varieties.

7. Contextual and Historical Remarks

Petrie matrices originated from convex geometry (“Petrie polygons”), but now pervade algebraic combinatorics, regular map theory, and equivariant topology. In every setting, their essence is the encoding of highly rigid local combinatorics—whether it is the zigzag structure of polygons, the algebraic independence in symmetric functions, or the weight compatibility in GKM graphs. Their determinantal values signal existence and uniqueness phenomena (e.g., unique tiling, maximal symmetry), thus serving as proxies for structural rigidity across disciplines.

The increasing sophistication of combinatorial proofs and graph-oriented interpretations suggests that Petrie matrices will remain central in the paper of symmetric function expansions, algebraic invariants, and symmetry phenomena in combinatorial and geometric contexts.