p-Stirling Numbers: Concepts & Applications

Updated 6 May 2026

p-Stirling numbers are generalized Stirling numbers that extend classical definitions through symmetric functions, explicit recurrences, and generating functions.
They offer concrete formulations including explicit formulas and combinatorial models, bridging operator calculus and p-adic analysis.
Applications span colored permutations, set partitions, and analytic continuations, making them vital for advanced combinatorics and number theory research.

A $p$ -Stirling number is a generalization of the classical Stirling numbers of both the first and second kind, parameterized by an integer $p$ and possessing deep combinatorial, algebraic, and $p$ -adic properties. These numbers admit distinct formal definitions depending on context, appearing in symmetric function theory, operator calculus, polynomial expansions, and $p$ -adic arithmetic. Below is a technical survey of their principal definitions, recurrence relations, generating functions, combinatorial models, $p$ -adic behavior, and applications.

1. Definitions and Fundamental Properties

The $p$ -Stirling numbers have two principal interpretations in the literature:

(a) Classical $p$ -Stirling numbers (parameter extension):

As defined by Merris, Sun, and formalized in the context of symmetric functions (Corcino et al., 2013), the $p$ -Stirling numbers of the first and second kind, denoted $c_p(n,k)$ and $S_p(n,k)$ , are given via symmetric function operations:

$p$ 0

where $p$ 1 and $p$ 2 denote the elementary and complete homogeneous symmetric functions, respectively.

These numbers satisfy the recursions:

$p$ 3

with boundary conditions $p$ 4, $p$ 5, and $p$ 6 (Corcino et al., 2013).

Explicit formula:

$p$ 7

(Corcino et al., 2013).

(b) Generalized (Parameter-Embedded) Stirling Numbers:

In the context of polynomial expansions and operator calculus, the term " $p$ 8-Stirling number" can also refer to the coefficients in the expansion of $p$ 9 (or more generally, products involving such powers) into the basis of falling factorials $p$ 0:

$p$ 1

where $p$ 2 is the classical $p$ 3-Stirling number of the second kind ( $p$ 4, $p$ 5) (Pita-Ruiz, 2018). The generalization allows arbitrary $p$ 6, $p$ 7, and higher multiplicities.

The explicit formula in this interpretation is:

$p$ 8

(Pita-Ruiz, 2018).

2. Generating Functions and Algebraic Structure

Both variants possess closed-form generating functions and inversion relations:

Type	Generating Function	Inverse Relation
$p$ 9	$p$ 0	$p$ 1
$p$ 2	$p$ 3	$p$ 4 and $p$ 5 inverse lower-triangular

Explicit formula for $p$ 6 above generalizes the Dobinski formula for ordinary Stirling numbers (Corcino et al., 2013, Pita-Ruiz, 2018).

3. Combinatorial and Algebraic Interpretations

Symmetric Function Model: The definitions via $p$ 7 and $p$ 8 relate $p$ 9-Stirling numbers to evaluations of symmetric functions at shifted integer sequences, connecting to the ring of symmetric polynomials (Corcino et al., 2013).
Colored Permutations and Set Partitions: $p$ 0 counts permutations of $p$ 1 into $p$ 2 disjoint cycles, where the distinguished $p$ 3 elements occur in separate cycles; $p$ 4 counts partitions of $p$ 5 into $p$ 6 blocks, each special block containing one of the distinguished $p$ 7 elements (Corcino et al., 2013).
Operator Calculus and Forest Models: In the context of operator expansions, e.g., $p$ 8, the coefficients generalize Stirling numbers with a bivariate $p$ 9-parameter and admit combinatorial interpretations via $p$ 0-forests and colored increasing trees (Mohammad-Noori, 2010).

4. $p$ 1-adic Properties and Asymptotic Behavior

The $p$ 2-Stirling numbers (classical or generalized) exhibit rich $p$ 3-adic behavior, notably:

$p$ 4-adic Valuations of Stirling Numbers: Explicit $p$ 5-adic valuation formulas exist for both $p$ 6 and $p$ 7 in certain congruence classes, such as $p$ 8, $p$ 9 (Adelberg, 2018, Adelberg et al., 2021). For instance,

$p$ 0

where $p$ 1 denotes the sum of the base- $p$ 2 digits [(Adelberg, 2018), Lemma 2.1]. In the case $p$ 3,

$p$ 4

where $p$ 5 is the number of $p$ 6's in the binary expansion of $p$ 7 (Adelberg, 2018).

$p$ 8-adic Analytic Continuation: For fixed $p$ 9, the sequence $p$ 0 admits a locally analytic $p$ 1-adic extension, with nontrivial congruence patterns arising from decomposition into $p$ 2-adic "balls" and controlled by the residue class of $p$ 3 modulo powers of $p$ 4 and $p$ 5 (Miska, 2018).
$p$ 6-adic Limits and Infinite Arrays: For suitable progressions in $p$ 7 and $p$ 8, $p$ 9 has a $c_p(n,k)$ 0-adic limit, defining $c_p(n,k)$ 1, with explicit closed forms when $c_p(n,k)$ 2 (Davis, 2013).

5. Generalizations and Applications

Generalized Stirling Numbers: The $c_p(n,k)$ 3-Stirling numbers fall within a much wider class of "generalized Stirling numbers" arising from the expansion of general polynomials $c_p(n,k)$ 4 into the falling factorial basis, with explicit recurrences and Dobinski-type formulas (Pita-Ruiz, 2018). Such numbers interpolate between the classical cases, $c_p(n,k)$ 5-Stirling numbers, and numerous other objects (e.g., degenerate, Lah, central factorial numbers) (Kim et al., 2022).
Convolution Relations, Orthogonality, and Determinants: There exist orthogonality relations:

$c_p(n,k)$ 6

convolution formulas relating parameters $c_p(n,k)$ 7, $c_p(n,k)$ 8 (e.g., $c_p(n,k)$ 9 in terms of $S_p(n,k)$ 0 and $S_p(n,k)$ 1), and determinantal expressions for block matrices built from $S_p(n,k)$ 2 (Corcino et al., 2013).

6. Connections with Polynomial Sequences and Umbral Calculus

The study of $S_p(n,k)$ 3-Stirling numbers is embedded in the wider project of associating generalized Stirling numbers with sequences of polynomials $S_p(n,k)$ 4, where the expansion

$S_p(n,k)$ 5

defines "P–Stirling numbers of the second kind" and a corresponding family of first-kind numbers inverts this relationship. Specializations of $S_p(n,k)$ 6 recover $S_p(n,k)$ 7-Stirling numbers, central factorial numbers, Bernoulli, Euler, and Lah numbers, and others (Kim et al., 2022).

Explicit generating functions, recurrences, and matrix inversion identities apply uniformly in the Sheffer–umbral framework:

$S_p(n,k)$ 8

where $S_p(n,k)$ 9 are Sheffer parameters for the sequence $p$ 00 (Kim et al., 2022).

7. Research Directions and Open Problems

Current research trends include:

Characterization of $p$ 01-adic valuation patterns and their stabilization, including fine-scale decompositions of $p$ 02-adic "balls" for Stirling number sequences (Miska, 2018, Adelberg et al., 2021).
Systematic investigation of $p$ 03-adic limits of combinatorial number arrays and the construction of analytic interpolants (Davis, 2013, Davis, 2014).
Elucidation of combinatorial interpretations for generalized and weighted Stirling numbers, especially via forest and colored permutation models (Mohammad-Noori, 2010, Corcino et al., 2013).
The role of higher-order Bernoulli numbers and their pole structure in determining $p$ 04-adic behavior (Adelberg, 2018).

A notable direction is the complete characterization of $p$ 05-adic valuation jumps in towers of the first-kind Stirling numbers, as partially confirmed for odd $p$ 06 by uniform "slope 2" phenomena (Hong et al., 2019), and the continued extension of umbral and operator methods to uncover new families of Stirling-like numbers.