Associated Stirling Numbers of the Second Kind

• Associated Stirling numbers of the second kind are defined as the number of ways to partition an n-element set into k blocks with a minimum block size of r, generalizing the classical Stirling numbers.
• They are studied using exponential generating functions, explicit composition-sum formulas, and recurrences that reveal deep combinatorial and probabilistic insights.
• Their applications extend to combinatorial enumeration, asymptotic analysis, and links with special functions such as poly-Bernoulli numbers and the Riemann zeta function.

The associated Stirling numbers of the second kind, also known as $r$-associated (or $m$-associated) Stirling numbers, enumerate set partitions with block-size lower bounds, generalizing the classical Stirling numbers which enumerate partitions with no restriction on block size. Recent developments have unified their combinatorial and analytic formulations, produced probabilistic representations, and established sharp asymptotic bounds. These numbers are fundamental in analytic combinatorics, algebraic enumeration, probability, and connections to special functions such as poly-Bernoulli and zeta values.

1. Definition and Fundamental Properties

For integers $n, k, r \geq 1$, the $r$-associated Stirling number of the second kind, denoted $S^{(r)}(n,k)$ (or equivalently in some references as $$\left\{\!\!\begin{smallmatrix} n \ k \end{smallmatrix}\!\!\right\}_{\geq r}$$), is the number of ways to partition an $n$-element set into $k$ nonempty blocks such that each block contains at least $r$ elements. Formally,

$$S^{(r)}(n,k) = \left|\{ \pi \vdash [n] : \pi \text{ has } k \text{ blocks},\, |\text{block}| \ge r\,\forall\text{ blocks} \}\right| \quad\text{for } n \ge rk,$$

with $S^{(1)}(n,k) = S(n,k)$, the classical Stirling numbers of the second kind. If $n < rk$ or $k < 0$, $S^{(r)}(n,k)=0$ (Wakhare, 2017, Komatsu et al., 2015).

Salient combinatorial interpretations include:

• $r=2$: partitions into blocks with no singletons ("associated" or "2-associated" Stirling numbers).
• $r=3$: partitions excluding blocks of size 1 or 2, etc.

2. Generating Functions and Explicit Formulas

The exponential generating function for the associated Stirling numbers is

$$\sum_{n=0}^{\infty} S^{(r)}(n,k)\,\frac{z^n}{n!} = \frac{1}{k!}\left(\sum_{s=r}^{\infty}\frac{z^s}{s!}\right)^{k}$$

for all $n \ge rk$ (Wakhare, 2017, Komatsu et al., 2015).

A more explicit version is given using truncated exponentials: $$\sum_{n=0}^{\infty} \left\{\!\!{n \atop k}\!\!\right\}_{\ge m}\frac{x^n}{n!} = \frac{1}{k!}\left(e^x - E_{m-1}(x)\right)^k,$$ where $E_{m-1}(x) = \sum_{j=0}^{m-1} \frac{x^j}{j!}$ (Komatsu et al., 2015).

An explicit composition-sum formula is

$$S^{(r)}(n,k)=\sum_{\substack{T_1+\cdots+T_k=n \ T_i\ge r}} \frac{n!}{k!\prod_{i=1}^{k} T_i!},$$

i.e., a sum over ordered $k$-tuples of positive integers at least $r$ summing to $n$ (Wakhare, 2017).

3. Recurrences and Structural Identities

Associated Stirling numbers of the second kind satisfy several fundamental recurrences. Given $n\ge 0$, $k\ge 1$, the elementary recurrence is

$$S^{(r)}(n+1,k) = \sum_{s=r}^{n+1} \binom{n}{s-1}\, S^{(r)}(n+1-s,k-1)$$

(Wakhare, 2017, Komatsu et al., 2015).

An alternative two-term recursion is

$$\left\{\!\!{n+1 \atop k}\!\!\right\}_{\ge m} = k \left\{\!\!{n \atop k}\!\!\right\}_{\ge m} + \binom{n}{m-1} \left\{\!\!{n-m+1 \atop k-1}\!\!\right\}_{\ge m}$$

(Komatsu et al., 2015).

These recurrences generalize the classical relations of $S(n+1,k) = k S(n,k) + S(n,k-1)$. Initial conditions are identical to the classical case aside from the $n\ge rk$ threshold.

4. Probabilistic Representations and Analytical Bounds

A key probabilistic development is the moment representation: $$S^{(r)}(n,k) = \frac{n!}{k! (r!)^k (n-rk)!}\,\mathbb{E}[M^{n-rk}],$$ where $M = X_1 + \cdots + X_k$, with $X_i$ i.i.d. Beta$(1, r)$ random variables ($g_r(x) = r(1-x)^{r-1},\, x\in[0,1]$) (Gismatullin et al., 13 Jan 2026). For $r=1$, this recovers the Irwin–Hall law for classical Stirling numbers.

Sharp upper and lower bounds, valid in distinct asymptotic regimes, follow from probabilistic inequalities:

Regime	Lower Bound	Upper Bound	Asymptotics
$k\to\infty$	$$\displaystyle \frac{n!}{k!(r!)^k(n-rk)!} \frac{k^{K}}{(r+1)^{K}}$$	See above, add $k^{K-1}$ correction term	Lower and upper asymptotically equal
$n\to\infty$	$\displaystyle S^{(r)}(n,k)\sim \frac{k^n}{k!}$	$\displaystyle S^{(r)}(n,k)\leq \frac{k^n}{k!}$	Equivalent to Poisson moment bounds
$r\to\infty$	Comparison with Erlang moments, exponential decay	High-$r$ behavior bounded by scaled exponential moments	Poisson/Erlang normalization emerges

For practical computation, the moment $\mathbb{E}[M^{n-rk}]$ can be evaluated via multinomial expansion, Monte Carlo, or numerical quadrature (Gismatullin et al., 13 Jan 2026).

5. Connections with Generalizations and Special Functions

Associated Stirling numbers occupy a central place in the broader landscape of combinatorial enumeration. Unified frameworks express partitions with block-size constraints via generating functions or potential polynomials $B(z)$ (Adell et al., 2024), with

$S_B(n,k) = [z^n] \frac{(B(z)-1)^k}{k!},$

and block-size sets $S$ (e.g., $S = \{r, r+1, \ldots\}$ for associated numbers) (Wakhare, 2017). All such families admit expansions, convolution recurrences, and inversion formulae parametrized by $S$ (Adell et al., 2024, Komatsu et al., 2015).

In analytic number theory, associated Stirling numbers appear in the formulation of incomplete/restricted poly-Bernoulli numbers. For parameter $\mu$, the incomplete poly-Bernoulli numbers are defined by

$$B^{(\mu)}_{n, \ge m} = \sum_{k=0}^n (-1)^{n-k} k! \left\{\!\!{n\atop k}\!\!\right\}_{\ge m} (k+1)^\mu,$$

with generating functions involving the polylogarithm and incomplete exponentials (Komatsu et al., 2015). For $m=2$, they underpin new series representations of the Riemann zeta function via the Lambert $W$ function.

6. Algebraic and Umbral Frameworks

Extensions through Sheffer sequences $P = \{p_n(x)\}$ provide a unifying algebraic viewpoint: $p_n(x) = \sum_{k=0}^n S_P(n,k)\, (x)_k,$ where $(x)_k$ is the falling factorial, and $S_P(n,k)$ gives the Stirling numbers of the second kind associated to $P$ (Kim et al., 2022, Adell et al., 2024). The exponential generating function in this context generalizes to

$$\sum_{n=k}^\infty S_P(n,k) \frac{t^n}{n!} = \frac{g(t)}{k!}(e^{f(t)} - 1)^k,$$

with $p_n(x)\sim (g(t), f(t))$. Specializations recover classical, associated, and degenerate Stirling numbers.

These matrices are invertible, and orthogonality relations linking first and second kinds persist in the associated case. Partition algebras, difference operators, and applications to moments of sums of i.i.d. variables are incorporated within this framework (Adell et al., 2024, Kim et al., 2022).

7. Applications and Further Directions

Associated Stirling numbers are vital for:

• Combinatorial enumeration: Enumerating block-restricted partitions, derangements, and generalizations.
• Asymptotic analysis: Normal and large-deviation limits, sharp bounds, and analytic combinatorics methods (Gismatullin et al., 13 Jan 2026).
• Special functions: Connections to Bell numbers, poly-Bernoulli numbers, and values of $\zeta(\mu)$ (Komatsu et al., 2015).
• Probabilistic analysis: Moment expansions for Poisson, Beta, or exponential sums. Probabilistic interpretations drive efficient approximations for large parameters (Gismatullin et al., 13 Jan 2026, Kim et al., 2017).
• Umbral and algebraic combinatorics: As generating functions for polynomial sequences, as transformation matrices, and in inversion/orthogonality identities (Kim et al., 2022, Adell et al., 2024).

These directions continue to motivate research in analytic, algebraic, and probabilistic combinatorics, as well as in applications to special function theory and mathematical statistics.