Hafnianinhos & Pfaffianinhos in Combinatorics

• Hafnianinhos and Pfaffianinhos are submatrix generalizations that extend classical hafnian and pfaffian functions, enabling detailed analysis of perfect and almost-perfect matchings.
• They are computed from principal submatrices and encode weighted matching statistics in monomer–dimer systems and quantum many-body wavefunctions.
• These constructs unify algebraic identities and computational techniques, offering new analytical tools for both combinatorial enumeration and quantum physics applications.

A Hafnianinho and Pfaffianinho refer to submatrix generalizations of the Hafnian and Pfaffian, which are algebraic objects central to combinatorial enumeration and quantum many-body wavefunctions. Hafnians and Pfaffians respectively encode the weighted sum over perfect matchings in symmetric and antisymmetric matrices. The Hafnianinho and Pfaffianinho constructions systematically record the matching statistics of principal submatrices, facilitating analysis of monomer-dimer systems, almost-perfect matchings, and the algebraic structure of topological wavefunctions.

1. Precise Definitions

Let $A = (A_{ij})$ be a symmetric $2L \times 2L$ matrix, and $K = (K_{ij})$ antisymmetric of the same size. The Hafnian is

$\haf(A) = \frac{1}{2^L L!} \sum_{i_1,\dots,i_{2L}=1}^{2L} \varepsilon^{i_1\cdots i_{2L}} A_{i_1 i_2} A_{i_3 i_4} \cdots A_{i_{2L-1} i_{2L}}$

which sums products of entries corresponding to all perfect matchings; $\varepsilon^{i_1\cdots i_{2L}}$ is the Levi–Civita symbol.

If $I = \{i_1,\dots,i_{2r}\} \subset \{1,\dots,2L\}$ with even cardinality, the Hafnianinho $\haf(A_{[I^c]})$ is the Hafnian of the principal submatrix on rows/columns $I^c$. For the Pfaffian,

$\pf(K) = \frac{1}{2^L L!} \sum_{i_1,\dots,i_{2L}=1}^{2L} \varepsilon^{i_1\cdots i_{2L}} K_{i_1 i_2} K_{i_3 i_4} \cdots K_{i_{2L-1} i_{2L}},$

and for $I$, the Pfaffianinho $\pf(K_{[I^c]})$ is the Pfaffian of the principal submatrix indexed by $I^c$, with sign correction $$\epsilon(I) = (-1)^{\sum_{s=1}^{2r}(i_s-1)}(-1)^{r(r-1)/2}$$ (Trino et al., 23 Nov 2025).

2. Combinatorial Interpretation

The Hafnian counts perfect matchings on undirected graphs with symmetric adjacency $A$; Pfaffians enumerate matchings for planar graphs with Kasteleyn orientation $K$. Hafnianinho and Pfaffianinho encode the number (and weighted sum) of perfect matchings on the induced subgraph when a set of vertices $I$ has been deleted (i.e., monomer “insertions”).

• Hafnianinho: $\haf(A_{[I^c]})$ counts perfect matchings in the subgraph on $I^c$.
• Pfaffianinho: $\pf(K_{[I^c]})$, up to sign, counts matchings under orientation-induced weighting and parity conventions (Trino et al., 23 Nov 2025).

3. Algebraic Identities and Mappings

A central theorem for planar graphs with Kasteleyn orientation $S$ and adjacency $A$ is

$\haf(A) = \pf(K),\qquad \haf(A_{[I^c]}) = \epsilon(I)\pf(K_{[I^c]})$

for any even $I$ (Trino et al., 23 Nov 2025).

The Jacobi identity for Pfaffians relates invertible $K$ and arbitrary $I$: $\pf((K^{-1})^T_{[I]}) = \frac{\epsilon(I)\pf(K_{[I^c]})}{\pf(K)},$ enabling algebraic translation between Hafnianinho and Pfaffianinho, and border-expansions

$\pf\!\begin{pmatrix}0 & u^T\ -u & K\end{pmatrix} = \sum_{j=1}^{2L}(-1)^{j+1}u_j\,\pf(K_{[\{j\}^c]})$

generating almost-perfect matchings with a single monomer (Trino et al., 23 Nov 2025).

4. Physical and Graph-Theoretic Applications

Hafnianinhos and Pfaffianinhos feature as correlation functions and partition sums in models of monomer–dimer systems and quantum Hall trial states. In monomer–dimer enumeration for planar graphs, the Monobisyzexant ($\Mbsz$) expansion expresses the partition sum as a direct sum over Pfaffianinhos

$\Mbsz(D,A) = \det(D) + \sum_{r=1}^{\lfloor L/2\rfloor} \sum_{I\subset[L],|I|=2r} \det(D_{[I]}) (-1)^r \pf(K'_I)\pf((K'_I)^{-1}_{[I]})$

where $K'_I$ is Kasteleyn matrix modified for defect lines for monomer sector $I$ (Trino et al., 23 Nov 2025).

In quantum Hall effect research, the Hafnian PH–Pfaffian state multiplies the PH–Pfaffian wavefunction by a Hafnian factor, mathematically equivalent to the compressed PH–Pfaffian state. These objects encode clustering, pairing, and topological sector information, and match correlation functions for higher moment pairing (Yang, 2022).

5. Berezin Integral Formalism and Computational Complexity

Hafnianinhos and Pfaffianinhos admit compact representations in terms of Berezin integrals over Gaussian Grassmann variables. This formalism gives efficient calculation methods for partition sums, correlation functions, and minor expansions in field-theoretic models. For planar graphs, Pfaffianinhos are computed in polynomial time ($O(L^3)$), while Hafnianinhos are $\#P$-complete in general and only tractable using sum identities on planar structures. Unitary block decomposition (for singular matrices) isolates null modes, yielding explicit sums over lower-dimensional Pfaffianinhos (Trino et al., 23 Nov 2025).

6. Illustrative Examples

Matrix Type	Object	Interpretation
$A$ (symmetric)	Hafnianinho	Matchings on $A_{[I^c]}$
$K$ (antisymmetric)	Pfaffianinho	Matchings on $K_{[I^c]}$
$K$ (planar Kasteleyn)	Pfaffian	Partition sum, $\pf(K) = \haf(A)$

For a $4\times4$ square adjacency $A$, $\haf(A) = 2$, and with $I=\{1,2\}$, $\haf(A_{[3,4]}) = 1$. For the Kasteleyn-oriented $K$, $\pf(K)=2$ and $\pf(K_{[3,4]})=1$, directly illustrating the correspondence.

7. Implications and Unifying Role

These submatrix generalizations unify the combinatorics of matchings, monomer–dimer systems, and almost-perfect matchings, supplying efficient computational techniques and analytic identities crucial for both condensed matter physics (e.g., wavefunction normalization, correlation statistics in quantum Hall states (Yang, 2022)) and combinatorial enumeration (planar graphs, spanning trees, monomer sector expansions (Trino et al., 23 Nov 2025)). The explicit mappings and unitary block strategies extend applicability to singular matrices and graphs of variable topology, positioning Hafnianinho and Pfaffianinho as central technical objects in modern graph-theoretic and physical models.