Petrie Symmetric Functions: Algebra & Combinatorics

• Petrie symmetric functions are defined by truncating exponents in monomials and interpolate between elementary and complete homogeneous functions.
• They expand in the Schur basis via determinantal Petrie matrices, resulting in signed multiplicity-free coefficients.
• They obey Pieri-like rules with combinatorial interpretations through k‑rim hooks and ribbon tilings that reveal underlying algebraic identities.

Petrie symmetric functions are a distinguished family of symmetric functions defined by truncating the range of exponents in the monomial basis, parameterized by a bound $k \in \mathbb{N}$ and degree $m \geq 0$. These functions interpolate fundamentally between the classical elementary symmetric functions $e_n$ and the complete homogeneous symmetric functions $h_n$. Their algebraic, combinatorial, and representation-theoretic properties are encoded by determinantal identities involving Petrie matrices. Central results include signed multiplicity-free Schur expansions, Pieri-like multiplication rules, algebraic independence properties, and combinatorial interpretations involving $k$-rim hooks and $k$-ribbon tilings.

1. Definition, Generating Functions, and Basic Properties

Petrie symmetric functions, denoted $G(k, m)$, are defined for fixed $k \geq 1$ and $m \geq 0$ as the sum of all degree-$m$ monomials in variables $x_1, x_2, \ldots$ where each exponent is strictly less than $k$:

$$G(k, m) = \sum_{\substack{|\alpha| = m\ \alpha_i < k\ \forall i}} x^{\alpha},$$

with $\alpha$ ranging over weak compositions of $m$.

The generating function for Petrie symmetric functions is

$$\sum_{m \geq 0} G(k, m) z^m = \prod_{i \geq 1} \left(1 + x_i z + x_i^2 z^2 + \cdots + x_i^{k-1} z^{k-1}\right) = \prod_{i \geq 1} \frac{1 - (x_i z)^k}{1 - x_i z}.$$

Special cases include:

• $k = 2$: $G(2, m) = e_m$, the $m$th elementary symmetric function.
• $k > m$: $G(k, m) = h_m$, the $m$th complete homogeneous symmetric function.
• $k = m$: $G(k, k) = h_k - p_k$, with $p_k$ the $k$th power sum symmetric function (Grinberg, 2020).

This interpolative nature underlines the modular character of the Petrie symmetric functions, making them a basis that unifies and extends classical families.

2. Schur Basis Expansion and Petrie Matrices

The expansion of Petrie symmetric functions $G(k, m)$ in the Schur basis is governed by the so-called $k$-Petrie numbers. Each coefficient in the expansion,

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

is a determinant evaluated on a Petrie matrix whose entries are $0$ or $1$ according to the inequality $0 \leq \lambda_i - i + j < k$:

$$\operatorname{pet}_k(\lambda, \mu) = \det\left([0 \leq \lambda_i - \mu_j - i + j < k]\right).$$

The Iverson bracket $[\cdot]$ equals $1$ when its condition holds and $0$ otherwise.

A classical theorem of Gordon–Wilkinson ensures that the determinant of any Petrie matrix is always $-1$, $0$, or $1$ (Grinberg, 2020, Cheng et al., 2022, Wu et al., 21 Sep 2025). Consequently, the Schur expansion of $G(k, m)$ is signed multiplicity-free: each $s_\lambda$ appears with coefficient $\pm1$ or $0$—a remarkable contrast to arbitrary nonnegative integer multiplicities found typically in symmetric function theory.

3. Pieri-like Rules and Ribbon Tiling Interpretations

Analogous to the classical Pieri rule for $h_n$ and dual Pieri rule for $e_n$, the Petrie symmetric functions satisfy a Pieri-like rule for multiplication with Schur functions. Given partition $\mu$,

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

with $\operatorname{pet}_k(\lambda, \mu)$ determined by the same determinantal formula. Grinberg's result is complemented by a combinatorial model: the coefficient is nonzero precisely when the skew shape $\lambda/\mu$ admits a unique proper $k$-ribbon tiling (with each ribbon—connected, $k$-cell rim hook without $2 \times 2$ blocks—left-justified in its row) (Jin et al., 2 Jun 2024, Wu et al., 21 Sep 2025). The sign of the coefficient is given by

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

where $r(\Theta_i)$ is the number of rows spanned by the $i$th $k$-ribbon. If more than one proper tiling exists, cancellations occur yielding zero. This refinement generalizes the classical rules to modular symmetric functions and corroborates the determinantal structure.

4. Combinatorial Models: Rim Hooks, Abaci, and Petrie Graphs

The combinatorial interpretation of the $k$-Petrie numbers involves $k$-rim hooks and the abacus model for partitions. For $\lambda$ with largest part $\lambda_1 < k$, the $k$-core structure and distinct rim hook removals fully control the expansion coefficient:

$$\operatorname{pet}_k(\lambda) = (-1)^{\sum_{i=1}^q \operatorname{ht}(\gamma_i) + q},$$

where $\gamma_1, \ldots, \gamma_q$ are the rim hooks of size $k$ successively removed, with height defined as one less than the number of rows occupied. The abacus—an array with beads positioned modulo $k$—encodes possible rim hook removals and reveals when the $k$-core is “small” (empty or single row). Maya diagram and orientation methods extend to skew shapes via Petrie graphs, whose good orientations relate directly to ribbon tilings and enable detailed generating function analysis (Cheng et al., 2022, Wu et al., 21 Sep 2025).

5. Algebraic Independence and Generating Sets

A notable algebraic property is that $\{G(k, 1), G(k, 2), \ldots\}$ forms an algebraically independent generating set for the algebra $\Lambda$ of symmetric functions over rings where $1-k$ is invertible. Thus, the Petrie symmetric functions serve as polynomial generators analogously to $h_n$ or $e_n$ (Grinberg, 2020). The proof relies on Hall inner products between power sums and Petrie functions, establishing the uniqueness of their determinantal coefficients.

6. Plethystic Pieri Rule and Power Sum Products

The plethystic Pieri rule—an extension of the Murnaghan–Nakayama rule—offers further combinatorial depth. For the plethysm $e_n \circ p_k$, the Schur expansion is a signed sum over all partitions obtained by adding $n$ left-justified $k$-ribbons to $\mu$:

$$(e_n \circ p_k) \cdot s_\mu = \sum_{\lambda} (-1)^{\sum_{i=1}^n \operatorname{ht}(\Theta_i)} s_\lambda,$$

with heights as above. This rule, proven by orientation-involution methods on Petrie graphs, connects generating function behaviors and symmetry properties for both $h_n \circ p_k$ and $e_n \circ p_k$ (Wu et al., 21 Sep 2025).

For products with power sums $p_n$, the expansion in the Schur basis is signed multiplicity-free if and only if $k \geq 3$, $k$ divides $n$, and $m \geq n$ (Cheng et al., 2022), confirming a conjecture of Alexandersson for $n=2$.

7. Connections to Generating Function Frameworks and Vertex Operators

The vertex operator realization framework for symmetric functions, as described in (Jing et al., 2016), provides a methodological blueprint for constructing generating functions—including those for Petrie symmetric functions—via choices of correlation function $f(x)$ and operators encoding creation and annihilation. While not treating Petrie functions directly, the framework allows their structure to be subsumed by adaptation of the Jacobi–Trudi approach with suitable modification of $f(x)$. This suggests deeper links between fermion-like operator relations and the modular determinantal rules governing Petrie symmetric functions.

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In summary, Petrie symmetric functions unify several strands—modular constructions, determinantal expansions, combinatorial tilings, and algebraic generation—in symmetric function theory. Their signed multiplicity-free Schur expansions, Pieri-like and plethystic rules, and refined connections to combinatorial and graphical models position them as a central object in algebraic combinatorics, with implications for representation theory and the structural analysis of symmetric function classes.