Bialternate Sum: Concepts & Applications

• Bialternate sum is a double alternation construction in algebra, combinatorics, and matrix theory that computes pairwise spectral sums and refined alternating statistics.
• It underpins matrix analysis by linking to the second additive compound and serves as a guardian map for detecting stability boundaries in control systems.
• The concept drives advances in partition theory and analytic number theory, offering tools for enumerative identities, convolution structures, and asymptotic error evaluation.

The bialternate sum refers to any of several constructions in combinatorics, algebra, and matrix theory that feature a double alternation—typically involving alternating signs or operations along two indices, dimensions, or parts. This concept appears in partition theory, the theory of multiplicative arithmetic functions, matrix analysis for control and stability, and in analytic number theory. In matrix theory, the bialternate sum is closely linked with the second additive compound, and in combinatorics and number theory, with refined alternating statistics, sum identities, and convolutional structures.

1. Definitions and Core Algebraic Structures

In matrix analysis, the bialternate sum (sometimes called the bialternate product) is an operation on matrices, particularly relevant for encoding spectral properties. For an $n\times n$ matrix $A$, the bialternate sum $A \diamondplus A$ is defined so that

$[A\diamondplus A]_{(p,q),(r,s)} = a_{pr}\delta_{qs} + a_{qs}\delta_{pr} - a_{ps}\delta_{qr} - a_{qr}\delta_{ps},$

where $(p,q)$ and $(r,s)$ index pairs with $p < q$, $r < s$, and $\delta_{ij}$ is the Kronecker delta (Bar-Shalom et al., 21 Sep 2025). This construction coincides with the second additive compound $A^{[2]}$: $$A^{[2]} := \left. \frac{d}{d\varepsilon} (I_n + \varepsilon A)^{(2)} \right|_{\varepsilon = 0}$$ where $(I_n + \varepsilon A)^{(2)}$ lists all $2\times2$ minors of $I_n + \varepsilon A$.

The spectrum of $A^{[2]}$ (and hence $A\diamondplus A$) consists of the pairwise sums $\lambda_i + \lambda_j$ for $i<j$, where $\lambda_i$ are the eigenvalues of $A$. In stability analysis, the determinant $\det(A^{[2]})$ serves as a guardian map—it vanishes precisely when $A$ has an eigenvalue on the imaginary axis, marking the boundary of Hurwitz stability for continuous-time systems (Bar-Shalom et al., 21 Sep 2025).

2. Bialternate Sums in Combinatorial and Partition Theory

In combinatorics, bialternate sum structures arise in the enumeration and classification of integer partitions, often as refinements tracking alternating statistics.

A classical instance is the refinement of the Lebesgue identity for partitions (Chen et al., 2010): $$\sum_k (–a q; q)_k / (q; q)_k \cdot q^{\binom{k+1}{2}} = (–a q^2; q^2)_\infty (–q; q)_\infty,$$ admitting a combinatorial bijection $\varphi$ that maps between partition pairs $(\alpha, \beta)$ and $(\mu, \nu)$, while preserving the difference of their alternating sum statistics. Specifically, for a partition $\lambda$,

$$|\lambda|_a = \lambda_1 - \lambda_2 + \lambda_3 - \lambda_4 + \ldots,$$

and the bijection satisfies $|\alpha|_a - n_o(\beta)$ is preserved, where $n_o(\beta)$ is the number of odd parts, and $|\mu|_a$ is the refined statistic on the "image" partition. The equality of counts for fixed values of the bialternate sum statistic is established: $$\#\left\{ (\alpha,\beta) : |\alpha|_a - n_o(\beta) = k \right\} = \#\left\{ (\mu,\nu) : |\mu|_a = k \right\}.$$ This "bialternate" property links different alternating features across the bijection, refining traditional combinatorial identities.

3. Analytic Number Theory: Multiplicative Functions and Generalized Alternation

Alternating and bialternate sums play a central role in analytic number theory, particularly in the paper of mean values and error terms for multiplicative arithmetic functions. For a classical multiplicative function $f(n)$,

$$S_1(x) = \sum_{n \le x} (-1)^{n-1} f(n), \quad S_2(x) = \sum_{n \le x} (-1)^{n-1} \frac{1}{f(n)}$$

are treated via a convolutional method involving Bell series expansion at the prime 2 and its reciprocal coefficients (Tóth, 2016). When generalizing the sign function to

$$t_Q(n) = \begin{cases} 1 & \text{if } q \not| n ~\forall q \in Q, \ -1 & \text{otherwise} \end{cases}$$

(where $Q$ is a set of primes), one obtains "bialternate" sums that alternate according to divisibility by each prime in $Q$. The corresponding Dirichlet series,

$D_Q(f,s) = \sum_{n=1}^\infty t_Q(n) f(n) n^{-s},$

can be systematically analyzed, and the approach extends to asymptotic evaluations and error bounds. These generalized sums are foundational for understanding cancellation and correlation phenomena in arithmetic functions.

One notable example is the alternating sum-of-divisors function $\beta(n)$ defined for prime powers by

$$\beta(p^a) = p^a - p^{a-1} + p^{a-2} - \cdots + (-1)^a,$$

and multiplicatively extended, with convolution identities exposing a "bialternate" character via double summation and duality properties (Tóth, 2011).

4. Matrix Stability: Spectral Criteria and Robustness via Bialternate Product

The bialternate product is a central tool in matrix stability analysis for robust Schur stability of uncertain systems. For a segment $C(\alpha) = \alpha A_1 + (1-\alpha) A_2$, robust stability is guaranteed if and only if two spectral conditions hold (Yilmaz, 27 Jun 2024):

• $(I-A_1)(I-A_2)^{-1}$ and $(I+A_1)(I+A_2)^{-1}$ have no negative real eigenvalues (ensuring no crossing through $\lambda = \pm1$).
• the companion-like matrix $M$ (constructed from the bialternate products of $A_1$ and $A_2$) has no real eigenvalues in $[1,\infty)$, ensuring no eigenvalue crosses the unit circle except at $\lambda = \pm1$.

These conditions yield an efficiently computable test for robust Schur stability across convex combinations and polytopes of matrices. The bialternate product reduces dimensionality, capturing crucial spectral behavior involving all pairwise products, relevant for discrete-time control systems.

5. Lie-Algebraic Guardian Maps and Stability Analysis

In the Lie-algebraic context, the bialternate sum (equivalently, the second additive compound) serves as a guardian representation (Bar-Shalom et al., 21 Sep 2025). Guardian maps are scalar-valued functions that vanish precisely when a system's matrix is on the boundary of stability (e.g., has an eigenvalue on the imaginary axis) and are nonzero otherwise. The guardian map property is associated with the determinant: $\det(A^{[2]}) = \det(A\diamondplus A)$ which is zero if any $\lambda_i + \lambda_j = 0$. By framing the bialternate sum and related constructions (e.g., Kronecker sum, lower Schläfli matrix) as Lie-algebra representations, a unifying principle is established for detecting stability boundaries across different algebraic settings.

Any similarity or contragredient representation of a guardian representation preserves the zero-locus property, making the approach robust under linear transformations and revealing the underlying algebraic invariance of stability detection mechanisms.

6. Applications in Combinatorics, Partition Theory, and Special Functions

Bialternate sums also feature in advanced combinatorial identities, especially in partition theory. For instance, refined alternating statistics as in the Lebesgue identity allow for sensitive enumeration tied to dual features of partitions (Chen et al., 2010). In hyperbolic Pascal triangles, the bialternate sum manifests via the synthesis of total and alternating sums with arbitrary weights: $$\tilde{s}_{(v,w),n} = \frac{v + w}{2} s_n^{(\text{total})} + \frac{v - w}{2} \tilde{s}_n^{(\text{alternating})},$$ generalizing classical alternation and connecting to broader invariants in geometric combinatorics (Németh et al., 2017).

In analytic and algebraic number theory, techniques for alternating (and thus bialternate) sums underpin explicit evaluations of Euler sums, multiple zeta values, and partition-related enumerations. Sign-reversing involutions, q-series manipulations, convolution identities, and generating function constructions are methodologically central for these evaluations (Bachraoui, 2016, Li, 2010).

7. Open Problems and Directions

Several open questions are highlighted in the literature concerning the behavior and deeper algebraic structure of bialternate sums:

• Classification and explicit evaluation of bialternate sums over various arithmetic functions.
• Improved asymptotic estimates and sharper error terms for bialternate sums in number-theoretic contexts (Tóth, 2016).
• Characterizations and combinatorial interpretations for positivity results in bialternate or truncated alternating sums of partition functions (Bachraoui, 2020).
• Extension of stability results to higher additive compounds and multi-index alternation for robust control in more complex matrix families (Yilmaz, 27 Jun 2024). These directions suggest ongoing research into both the structural properties and computational aspects arising from double alternation across mathematical disciplines.

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In summary, the bialternate sum is a multifaceted construction pervasive in combinatorics, arithmetic, matrix analysis, and algebraic stability theory. By encapsulating double alternation—either as spectral pairwise operations, refined combinatorial statistics, or dual convolutional analogs—it facilitates fine-grained analysis of stability, enumeration, and symmetry properties that are invisible to single alternation or unlayered sum structures. The equivalence with additive compounds and the Lie-algebraic perspective point toward its foundational role in modern mathematical analysis and applications in control, number theory, and combinatorics.