Graphical Stirling Numbers

• Graphical Stirling numbers extend classical Stirling numbers by counting structured partitions in graphs, such as independent sets, cycles, and forests.
• They offer a unified combinatorial framework that connects graph coloring, rook theory, and homological invariants in algebraic topology.
• Analytical approaches reveal real-rootedness, asymptotic normality, and deep links to normal ordering in noncommutative algebras, enhancing enumerative techniques.

Graphical Stirling numbers generalize classical Stirling numbers by encoding combinatorial invariants of graphs, notably counting structured partitions such as independent sets, cyclic decompositions, or forests, with profound connections to normal ordering in noncommutative algebras, graph coloring, homological algebra, and combinatorial representation theory. Both the first and second kind admit graph-dependent formulations, often leading to rich enumerative and algebraic phenomena relevant across combinatorics, algebraic topology, and mathematical physics.

1. Classical and Graphical Formulations

Classical Stirling numbers of the first kind, $s(n,k)$, enumerate permutations of $n$ elements with exactly $k$ cycles, while those of the second kind, $S(n,k)$, count partitions of $n$ elements into $k$ nonempty blocks. Graphical Stirling numbers extend these by considering a graph $G=(V,E)$ and counting structural partitions of $V$ dictated by the graph’s edge set. Specifically:

• The graphical Stirling number of the second kind, $S(G,k)$, is the number of partitions of $V$ into $k$ nonempty independent sets, equivalently the number of proper $k$-colorings using all $k$ colors up to permutation of colors (Thanh et al., 2012, Gonzales, 2021).
• Graphical Stirling numbers of the first kind, $c(G,k)$, enumerate $k$-cycle decompositions of directed graphs subject to adjacency constraints (Gonzales, 2021). These definitions specialize to the classical forms when $G$ is empty.

2. Homological Interpretation: Stirling Complexes and Genus-1 Graph Homology

In genus-1 commutative graph homology with $n \ge 3$ markings, the homology $H_*(G_{1,n})$ decomposes as a direct sum over complexes $S_{n,k}$ ("Stirling complexes") indexed by $k=1,\ldots,n$, where each complex is the chain complex of decorated trees ("Stirling trees") (Ward, 2023). The key properties are:

• Each $S_{n,k}$ has homology concentrated in the top degree, $\dim_{\Q} H_*(S_{n,k}) = |s(n,k)|$, matching the unsigned Stirling numbers of the first kind.
• The differential on $S_{n,k}$ encodes both internal-edge collapses and combinatorial operations on distinguished sets of flags.
• The overall homology has rank $\sum_{k=1}^n |s(n,k)| = n!$, only nonzero for $k$ matching the parity of $n$.

This refinement realizes each $s(n,k)$ as a Betti number and embeds the symmetric group representation theory into graph homology. The construction leverages Koszul duality, operadic bar complexes, and set partitions of input flags.

3. Enumeration via Partitioning and Coloring: Forests, Cycles, Multipartite Graphs

Graphical Stirling numbers admit explicit enumeration for several graph families:

• For forests with $c$ components, $$S(F_n^c, k) = \sum_{i=0}^{c-1} \binom{c-1}{i} S(n-1-i, k-1)$$, revealing Stirling polynomial generating functions with real-rootedness and interlacing properties (Thanh et al., 2012).
• For complete $\ell$-partite graphs $K(n_1,\dots,n_\ell)$, $S(G,k)$ is computed as $$\sum_{j_1+\dots+j_\ell=k,\, j_i\ge1} \prod_{i=1}^\ell S_2(n_i, j_i)$$, with the graphical Bell number $B(G) = \prod_{i=1}^\ell B_{n_i}$ (Allagan et al., 7 Dec 2025).
• Specialization to multipartite, bipartite with removed matchings, and Mycielskian trees yields formulas linking $S(G,k)$ and $B(G)$ to sums over Bell numbers and classical combinatorial sequences (OEIS entries A000051, A096376, etc.). These enumerative results demonstrate how graphical Stirling numbers model the complexity of partitioning in constrained graph settings.

4. Graphical Stirling Numbers, Rook Theory, and Normal Ordering

Many graphical Stirling numbers are intimately tied to normal ordering in the Weyl algebra. For a word $w$ (with prescribed $x$ and $D$ occurrences), the normal-ordered expansion coefficients $S_w(k)$ correspond to partitions of the associated quasi-threshold graph $G_w$ into $k$ independent sets (Engbers et al., 2013, Gonzales, 2021, Eu et al., 2017). Important structural features:

• Rook placements on Ferrers boards derived from Dyck words enumerate respective graphical Stirling numbers; file placements encode first-kind analogs (Gonzales, 2021, Eu et al., 2017).
• Lah numbers and corresponding graphical analogs enumerate ordered partitions into lists avoiding graph edges.
• $q$-analogues arise naturally via weighting placements or partitions by inversion or adjacency statistics.
• Chromatic polynomial expansions and inclusion-exclusion formulas relate $S(G,k)$ to $X_G(x)$.

This perspective unifies algebraic combinatorics and graph theory, yielding powerful enumeration algorithms and revealing deep connections to operator algebra.

5. Structural Properties and Asymptotics

Graphical Stirling polynomials $\sigma(G,x)=\sum_k S(G,k)x^k$ inherit striking analytic properties:

• For forests and cycles, generating functions have all real and nonpositive zeros; interlacing between zeros for sequential graphs is established.
• Asymptotic normality holds: For a random proper coloring of a forest or cycle sampled uniformly among those with $k$ classes, the normalized class number converges in distribution to $N(0,1)$ under mild growth of graph parameters (Thanh et al., 2012).
• Log-concavity and unimodality are conjectured and often empirically verified for sequences $k\mapsto S_{m,m}(n,k)$ in block colorings of multipartite graphs and for graphical Stirling numbers more broadly (Allagan et al., 7 Dec 2025, Codara et al., 2013).

These results show that graphical Stirling numbers retain, and frequently enhance, the robust combinatorial and probabilistic behavior of their classical counterparts.

6. Conjectured Identities, Graph Homology, and Open Problems

Recent research proposes algebraic identities and conjectures for graphical Stirling numbers involving weighted partitions and monomer-dimer models on regular graphs. For given integers $g$ and $w$ ($0\le w\le g-2$) and $g$ distinct parameters, the sum over ordered partitions weighted by products of binomial terms vanishes, connecting Stirling numbers, graph matchings, and cycle decompositions (Federbush, 2018): $$\sum_{\text{weighted configurations}} (-1)^r \prod_{i=1}^r P_{w_i}(t_i) = 0,$$ where $P_w(x)$ is the falling factorial polynomial and $t_i$ is the sum of parameters in block $S_i$.

These conjectures, verified computationally for small parameters, reside at the intersection of exponential generating functions, matchings in graphs, and advanced combinatorial identities. Establishing general proofs and unraveling further structural meaning remains open.

7. Graphical Stirling Numbers in Algebraic and Combinatorial Contexts

The graphical theory of Stirling numbers provides homological, representation-theoretic, and combinatorial refinement:

• In graph homology, Betti numbers of tree complexes derived from decorated combinatorial objects realize Stirling numbers of the first kind (Ward, 2023).
• Representation-theoretic models via symmetric group actions and FI-modules are enabled by the explicit Sn-equivariant decompositions.
• Categorification processes transform classical combinatorial identities into exact chain-level statements in graph complexes, indicating the deep structural role of graphical Stirling numbers in modern mathematical frameworks.

This theory links diverse domains including algebraic topology, operad theory, symmetric function theory, and statistical mechanics, and suggests graphical Stirling numbers will continue to uncover structural insights in combinatorics and beyond.