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Arithmetic Properties of Mixed Stirling Numbers of the second kind

This paper investigates the arithmetic properties of mixed Stirling numbers of the second kind by establishing their recurrence relations and generating functions, analyzing their behavior modulo $p$ and $p^2$ , and extending the classical Touchard congruence to reveal unique number-theoretic signatures distinct from classical set partitions.

Original authors: Daniel Yaqubi, Madjid Mirzavaziri

Published 2026-08-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel Yaqubi, Madjid Mirzavaziri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are at a massive party where everyone wants to form groups. In the world of mathematics, there's a classic game called "Stirling numbers of the second kind." It answers a simple question: If you have a bunch of distinct guests, how many different ways can you split them into non-empty groups? It's like counting the possible seating arrangements for a dinner party where the order of the guests at the table doesn't matter, but who sits with whom does.

Now, imagine the party gets a little more complicated. Some guests are wearing name tags (labeled), while others are just anonymous faces (unlabeled). Maybe some tables are distinct because they have different colors, while others are identical. This is the world of "mixed partitions." Mathematicians call these arrangements "mixed Stirling numbers." They are counting how many ways you can organize your guests when you have a mix of labeled and unlabeled groups.

Why does anyone care about counting party arrangements? It turns out these numbers are like the DNA of counting problems. They show up everywhere in computer science, probability, and even in understanding how numbers behave when you divide them by primes (like 2, 3, 5, 7). If you look at these numbers through the lens of "modular arithmetic"—which is basically just looking at the remainders after division—they reveal hidden patterns and rhythms, almost like a secret code. Understanding these patterns helps mathematicians predict how complex systems behave, from cryptography to the structure of the universe.

The Paper's Story: Cracking the Code of Mixed Parties

In this paper, authors Daniel Yaqubi and Madjid Mirzavaziri decide to take a deep dive into these "mixed Stirling numbers." They aren't just counting the parties; they are investigating the arithmetic secrets hidden inside the numbers, specifically looking at what happens when you divide these counts by a prime number $p$ or its square $p^2$ . Think of it as checking if the number of ways to arrange the party guests leaves a specific "remainder" when you count them in groups of 7, or 49, or 121.

The authors start by building a solid foundation. They prove that these mixed numbers follow a specific set of rules, called "recurrence relations." Imagine you have a party with $n$ guests. If you add one more guest, the number of ways to arrange the party isn't random; it's directly related to how you could have arranged the party with $n-1$ guests. The paper writes down the exact formula for this relationship, showing how the "labeled" and "unlabeled" parts of the party interact. They also provide a "generating function," which is like a magical machine that, when you feed it a variable, spits out all the possible counts for every party size at once.

The real magic, however, happens when they look at these numbers modulo a prime number. The authors extend a famous old rule called the "Touchard congruence." For a long time, mathematicians knew that for standard party arrangements (where all groups are unlabeled), the number of ways to arrange $n+p$ guests is related to the number of ways to arrange $n$ and $n+1$ guests when you look at the remainders after dividing by $p$ . It's a beautiful, predictable rhythm.

Yaqubi and Mirzavaziri show that this rhythm doesn't disappear just because we mix in labeled and unlabeled groups. They prove that for their "mixed" parties, a similar rule holds true. If you have a prime number $p$ , and you look at the number of ways to arrange $p+n$ guests, it is congruent (meaning it leaves the same remainder) to the number of ways to arrange $n+1$ guests, provided the number of groups isn't too large compared to $p$ . They use a clever mathematical tool called the "Frobenius differential operator" to show this. You can think of this operator as a special kind of microscope that zooms in on the structure of the numbers and reveals that the "labeled" and "unlabeled" parts dance together in a way that preserves the old rhythm.

The paper goes even deeper, looking at what happens when you divide by $p^2$ (the square of the prime). This is like checking the remainder not just for groups of 7, but for groups of 49. Here, the authors find that the mixed numbers have a very specific "signature." They show that these numbers are divisible by $p$ , and they give a precise formula for what the remainder is when divided by $p^2$ . This remainder isn't random; it's connected to other famous mathematical constants like Bernoulli numbers and Fermat quotients. It's as if the party arrangement count is whispering a secret about the fundamental nature of prime numbers.

One of the most exciting findings is that these mixed numbers behave with "p-adic continuity." This is a fancy way of saying that if you change the number of guests by a certain amount related to the prime $p$ , the remainder of the count doesn't jump around wildly; it stays smooth and predictable. The authors prove that these numbers can be extended into a continuous function, meaning the discrete jumps between party sizes actually form a smooth curve when viewed through the right mathematical lens.

The paper also touches on the simplest case: what happens when you divide by 2? They show that the parity (whether the number is even or odd) of these mixed arrangements depends entirely on how many labeled groups you have. If you have three or more labeled groups, the number of arrangements is always even. If you have fewer, it follows a pattern based on binomial coefficients, which are the numbers you see in Pascal's Triangle.

Finally, the authors look ahead. They define new "Mixed Bell numbers," which are the total sum of all possible mixed arrangements for a given number of guests. They provide the formulas for these new numbers and suggest that they likely follow similar rhythmic patterns to the ones they just discovered. They don't claim to have solved everything; instead, they open the door for future researchers to explore these new families of numbers, asking if they too hold the secrets of the primes.

In short, this paper takes a complex, hybrid version of a classic counting problem and shows that it still sings the same mathematical song as its simpler cousin. By proving that these mixed arrangements follow predictable rules when divided by primes, the authors have added a new, vibrant chapter to the story of how numbers organize themselves. They haven't just counted the parties; they've discovered the music the parties are dancing to.

Technical Summary: Arithmetic Properties of Mixed Stirling Numbers of the Second Kind

Problem Statement
The paper investigates the arithmetic properties of the "mixed Stirling numbers of the second kind," denoted $S(n; c)$ . These numbers generalize the classical Stirling numbers of the second kind, $\left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\}$ , which count partitions of $n$ distinct elements into $k$ non-empty, unlabeled blocks. The authors extend this to a "mixed" occupancy problem where the target blocks consist of two types: $m$ unlabeled non-empty blocks and $k$ labeled non-empty blocks. This configuration is encoded by the vector $c = (m, 1^k)$ . The primary objective is to analyze the behavior of these numbers modulo a prime $p$ and its higher powers ( $p^2, p^3$ ), specifically seeking to extend classical results such as the Touchard congruence and Kummer-type congruences to this mixed framework.

Methodology
The authors employ a combination of combinatorial analysis, generating functions, and $p$ -adic number theory.

Combinatorial Derivations: Fundamental recurrence relations and explicit formulas are derived by considering the placement of the $n$ -th element into existing or new blocks. The authors utilize the Principle of Inclusion-Exclusion to establish explicit formulas involving power sequences and binomial coefficients.
Generating Functions: The exponential generating functions (EGF) for these numbers are established. For the configuration $c = (m, 1^k)$ , the EGF is identified as $\Phi_{m,k}(x) = \frac{(e^x - 1)^{m+k}}{m!}$ . This form allows the authors to leverage the properties of the exponential function in modular arithmetic.
$p$ -adic Analysis: The core of the arithmetic investigation relies on classical tools:
- Legendre's Formula and Kummer's Theorem are used to determine $p$ -adic valuations of factorials and binomial coefficients.
- Fermat Quotients ( $q_p(j)$ ) and Second Fermat Quotients ( $q_p^{(2)}(j)$ ) are utilized to lift congruences from modulo $p$ to $p^2$ and $p^3$ .
- Frobenius Differential Operators: The authors apply the operator $D^p \equiv D \pmod p$ to the generating functions to derive periodicity results.
- Bernoulli Numbers: The results are linked to classical Bernoulli numbers ( $B_n$ ) and their congruences, particularly the Kummer congruences.

Key Contributions and Results

Fundamental Properties: The paper establishes recurrence relations (Theorem 2.1) and explicit formulas (Theorem 2.3) for $S(1^n; c)$ . It demonstrates that these numbers can be expressed as linear combinations of power sequences, facilitating $p$ -adic interpolation. The symmetry of the numbers with respect to permutations of the cell multiplicity vector is also formalized (Proposition 2.5, Lemma 2.6).
$p$ -adic Valuations: The authors derive lower bounds for the $p$ -adic valuation of mixed Stirling numbers (Theorem 3.1, Theorem 3.2). These bounds reduce the problem to the valuations of classical Stirling numbers and binomial coefficients, showing that mixed numbers inherit divisibility properties from their classical counterparts.
Congruences Modulo $p$ and $p^2$ :
- Touchard-Type Congruence: The paper extends the classical Touchard congruence ( $B_{n+p} \equiv B_{n+1} + B_n \pmod p$ ) to the mixed framework. Theorem 3.19 proves that $S(p+n; (m, 1^k)) \equiv S(n+1; (m, 1^k)) \pmod p$ , provided $m+k < p$ . This is attributed to the invariance of the generating function under the Frobenius differential operator.
- Mirimanoff-Type Congruences: For $n=p$ , the authors derive specific congruences modulo $p^2$ (Theorem 3.5). These results express $S(p; c)$ in terms of Stirling numbers of the second kind and binomial coefficients, and further relate them to Bernoulli numbers (Eq. 11).
- Specific Cases: Corollary 3.6 provides explicit residues for specific configurations (e.g., $k=2$ and $k=3$ ), linking the results to Wolstenholme-type and Mirimanoff-type congruences. Notably, the case $k=2$ is connected to the arithmetic of Wieferich primes via the Fermat quotient of 2.
Higher Order Congruences and Continuity:
- Theorem 3.14 and Corollary 3.15 extend the analysis to modulo $p^3$ , expressing the numbers in terms of first and second Fermat quotients and generalized Bernoulli numbers.
- Conditional $p$ -adic Continuity: Lemma 3.12 establishes the $p$ -adic continuity of the sequence $S(n; c)$ under a specific hypothesis. It proves that if the classical Stirling numbers satisfy the congruence $\left\{ \begin{smallmatrix} N + p^a(p-1) \\ k \end{smallmatrix} \right\} \equiv \left\{ \begin{smallmatrix} N \\ k \end{smallmatrix} \right\} \pmod {p^a}$ for all sufficiently large $N$ , then the mixed Stirling numbers satisfy $S(n + p^a(p-1); c) \equiv S(n; c) \pmod {p^a}$ for sufficiently large $n$ .
Parity Analysis: Proposition 3.18 characterizes the behavior of these numbers modulo 2, showing that for $k \ge 3$ , the numbers vanish, while for $k=1, 2$ , they follow specific binomial parity patterns related to the Sierpiński gasket.

Significance and Claims
The paper claims that mixed Stirling numbers possess "unique number-theoretic signatures distinct from classical set partitions," yet they retain deep structural connections to classical sequences. The primary significance lies in the successful extension of the Touchard congruence and the derivation of Mirimanoff-type congruences for this mixed framework.

The authors argue that the inclusion of unlabeled blocks (scaling by $1/m!$ ) interacts with $p$ -adic valuations in a way that preserves the underlying Frobenius periodicity of the exponential generating function, provided the total number of blocks is less than $p$ . This suggests a broader principle: combinatorial sequences with exponential generating functions constructed from powers of $e^x - 1$ with $p$ -integral coefficients naturally exhibit Touchard-type congruences.

Furthermore, the work establishes an explicit arithmetic bridge between mixed partitions and the theory of Fermat quotients and Bernoulli numbers. The results indicate that the arithmetic of mixed Stirling numbers acts as a "combinatorial mirror" to the local structure of $p$ -adic quantities, such as the Fermat quotient $q_p(2)$ and the values of the $p$ -adic zeta function.

Future Directions
The paper concludes by defining new families of "mixed Bell numbers" ( $MB_{n,k}$ and $TB_n$ ) that sum over the unlabeled block counts. The authors identify the analysis of the modular behavior of these new sequences as an open problem, noting that their generating functions contain non-trivial denominators and singularities that may yield distinct arithmetic signatures warranting further exploration.

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