CountGD++: Advanced Counting Algorithms

• CountGD++ is a family of advanced, domain-specific counting algorithms that use dynamic programming and recurrence relations to achieve constant-factor improvements in computational efficiency and exact counts.
• It demonstrates significant optimizations in enumerating structures such as graphical degree sequences, phylogenetic networks, LDPC code cycles, and Gray-code patterns through tailored recurrences and memory-saving techniques.
• CountGD++ integrates methods from prompt engineering in computer vision with combinatorial tools from graph theory, coding theory, and algebra, offering a versatile toolkit for both theoretical and applied research.

CountGD++ designates a family of advanced, domain-specific counting algorithms that implement improved combinatorial enumeration for structures such as graphical degree sequences, phylogenetic networks, cycles in @@@@1@@@@ codes, Gray-code matrix patterns, and gcd-pairs in finite groups. Across all domains, CountGD++ leverages problem-specific dynamic programming, recurrence relations, or prompt engineering to achieve constant-factor improvements in computational efficiency, exactness of counts, and extension to broader parameter regimes. The methodologies and optimizations are rigorously documented in the literature spanning graph theory, coding theory, computational algebra, and computer vision.

1. Graphical Degree Sequence Enumeration

CountGD++ for zero-free graphical degree sequences, as developed by Wang (Wang et al., 2018), centers on the recursive computation of the Barnes–Savage function $P(N,k,l,s)$. This function counts the integer partitions of $N$ into at most $l$ parts, each at most $k$, such that initial segments satisfy corank-sum inequalities corresponding to the Erdős–Gallai characterization. The algorithm maintains a four-dimensional array $Q[\ell][k][N][s]$, optimized via:

• Mod-$2$ reuse for the $\ell$ dimension (storing only two layers at a time)
• Ragged allocation exploiting tight upper and lower bounds for $N$ and $s$
• Skipping non-viable ranges for $s$ based on explicit bounds
• In-place updates and immediate summing to produce cumulative values for all sequence lengths up to $n$

Complexity analysis demonstrates $O(n^6)$ time and $O(n^5)$ space, but empirically yields a $10\times$ speed-up (constant-factor) over prior methods. Extensions include counting degree sequences allowing zeros, $k$-connected degree sequences, and classes of partitions via modified summations.

Parameter	Original DP	CountGD++ DP
Space	$O(n^5)$	$\approx 0.1 \times O(n^5)$
Time	$O(n^6)$	$\approx 0.1 \times O(n^6)$

CountGD++ thus serves as the definitive dynamic programming method for enumerating graphical partitions and related combinatorial objects in graph theory.

2. Galled Network Enumeration in Phylogenetics

CountGD++ for phylogenetic networks (Gunawan et al., 2018) employs a combinatorial correspondence between "1-galled networks" and twin-cherry-free dup-trees. The core recurrence relation for the number $N^{(i)}_{k}$ of dup-trees with $i$ duplications over $k$ taxa is:

$$N^{(i+1)}_k = (k+i-2) N^{(i)}_k + iN^{(i-1)}_k + \frac{1}{2} \sum_{d=1}^{i} \binom{i}{d}(2d-1)!!(N^{(i-d)}_{k-d} - N^{(i-d)}_{k-d+1})$$

Global counts for 1-galled ($G_1(k)$) and general galled networks ($G(k)$) are assembled via aggregation over rooted tree decompositions, using product formulas on guide trees ($A_k$). CountGD++ implements recursion with precomputed double-factorials and binomial coefficients, realizing $O(K^3)$ time and $O(K^2)$ space. For tractable $k$ ($\lesssim 12$), full enumeration is possible.

Network Class	Enumeration Method	Complexity
1-galled	Dup-tree recurrence	$O(K^3)$
General galled	Product over trees	Superexp. in $k$

This approach provides the state-of-the-art for exact counting and enumeration of recombination networks in evolutionary biology.

3. Cycle Counting in QC and APM LDPC Codes

CountGD++ in coding theory (Gholami et al., 2023) delivers efficient enumeration of cycles in the Tanner graphs of quasi-cyclic (QC) and affine permutation matrix (APM) LDPC codes. The algorithm reduces global cycle counting to enumerating tailless backtrackless closed (TBC) walks of fixed length $L$ in the protograph $G_b$. Key algorithmic steps:

• Enumerate all non-isomorphic TBC walks via depth-limited DFS on $G_b$
• For each walk, compute its period $n(W)$ under group shifts, which determines multiplicity in the lifted graph
• Directly count cycles as $N_L = \sum_{W\in\mathcal{A}} m / n(W)$, independent of the large circulant parameter $m$

Complexity depends solely on base matrix parameters ($k$, $b_{\max}$, $L$) and remains polynomial for moderate $L$, in contrast to prior adjacency matrix and message-passing methods whose cost scaled exponentially with $m$.

LDPC Cycle Type	Previous Cost	CountGD++ Cost
QC cycles	$\gg O(N^{[L/2]})$	$O(k^{2L}b_{\max}^{2L})$
APM cycles	Similar, plus parameter tests	Same

CountGD++ thus establishes an optimal regime for high-throughput cycle spectrum computation in code design.

4. Gray-Code Pattern Counting in Discrete Geometry

CountGD++ as employed in geometric combinatorics (Kliem, 2021) classifies $k$-bit Gray code patterns modulo 2, crucial for equipartition problems such as the Grünbaum–Hadwiger–Ramos problem. The method:

• Counts equiparting matrices built from blockwise concatenated Gray codes
• Applies parity criteria under group actions $(\mathbb{Z}/2)^k \rtimes S_k$ to determine which transition-count vectors yield odd numbers of Gray code realizations
• Determines exact bounds for equipartition, leading to improvements such as $d \ge 2^n(1+2^{k-1})$ for hyperplane partitioning of measures

Algorithmic counting uses delta-sequence enumeration, dyadic-valuation pruning, and orbit-size calculations, attaining $O(k! k)$ complexity for the parity tests involved.

Equipartition Bound	Parity Classifier	Algorithmic Cost
Mani-Levitska–Vrećica	$(\mathbb{Z}/2)^k$	$O(k! k)$
Blagojević et al. 2016	$(\mathbb{Z}/2)^k \rtimes S_k$	$O(k! k)$

CountGD++ thus mechanizes exact enumeration for topological and combinatorial bounds in discrete geometry.

5. Integrated Random Walks for Graphic Sequence Counting

CountGD++ in the context of graphical degree sequences via random walks (Balister et al., 2023) reformulates the enumeration as counting symmetric random walk bridges with area (integral) constraints. The method:

• Maps degree sequence constraints to persistence events for integrated lazy SSRW bridges
• Computes the DP function $F(N,y,a)$ representing the number of walks of length $N=n-1$ ending at heights $y$, with area $a$, subject to non-negativity and parity
• Exploits optimized DP recurrence:

$$F(N,y,a) = F(N-1,y+1,a+y+1) + F(N-1,y-1,a+y-1) + 2 \cdot F(N-1,y,a+y)$$

• Achieves $O(n^3)$ time and space via compressed array representations and boundary pruning

Exact counts $G(n)$ extend to $n \le 1651$ and confirm the refined asymptotic $G(n) = (c+o(1)) 4^n / n^{3/4}$.

$n$	$G(n)$
1	1
2	2
8	120
15	15493

This approach replaces enumeration of degree sequences with efficient DP on lattice paths, drastically improving computational reach.

6. GCD-Pair Enumeration in Finite Groups

For counting unordered gcd-pairs in $\mathbb{Z}_n$ (Tapanyo et al., 2022), CountGD++ utilizes an explicit divisor-sum formula:

$$|\nu_n| = \sum_{d\mid n} \left(1 + \sum_{k=1}^{n/d - 1} \phi(k) \right) - 1$$

where $\phi(k)$ is Euler's totient function. Optimized algorithms include:

• Precomputing $\phi[1..n-1]$ via sieve
• Building prefix sums for rapid partial evaluations
• Looping over divisors using $O(\sqrt{n})$ factoring

The method enables immediate generalization to subset counts (units, zero-divisors) and supports efficient implementations for large $n$.

$n$	$|\nu_n|$
6	16
12	64
30	388

CountGD++ thus formalizes divisor-structured enumeration in computational algebra.

7. Generalized Prompting for Open-World Counting in Computer Vision

CountGD++ for vision counting (Amini-Naieni et al., 29 Dec 2025) defines a multi-modal model that incorporates generalized prompting:

• Accepts open-ended text and visual exemplars for "what to count" and "what not to count," expanding beyond prior methods which restricted negative prompts
• Implements pseudo-exemplar generation: model selects top candidate regions to improve prompt specification iteratively
• Utilizes attention mechanisms across image and prompt modalities, and transformer-based object query heads (Swin-Transformer + BERT)
• Supports direct integration as a "vision expert" agent within LLM pipelines

Benchmark evaluations across FSCD-147, PrACo, ShanghaiTech, and other datasets demonstrate state-of-the-art accuracy with substantial improvements in F-score, RMSE, and AP metrics over prior models.

Dataset	Metric	CountGD++	Baseline
FSCD-147	RMSE (text only)	27.03	82.99 (CountSE)
Blood Cell	MAE (pos+neg)	1.52	10.99 (CountGD)
VideoCount	MAE	12.3	69.1 (text only)

The architecture, training regime, and agent loop allow robust, generalizable object counting and detection in open-world and controlled settings.

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CountGD++ interventions typify problem-tailored combinatorial enumeration, algorithmic innovation, and computational efficiency across disparate mathematical, computational, and applied domains. The unifying theme is the formal reduction of global counting to tractable, structured recurrences, leveraging problem symmetries and exactness, and extending to integration with modern ML frameworks.