CountOCC: Structured Counting Algorithms

• CountOCC is a framework combining methods for counting structured objects across diverse domains including computational geometry, combinatorial sequences, and permutation patterns.
• It employs optimized data structures, recurrence relations, and dynamic programming to improve space–time tradeoffs and achieve robust performance.
• Applications span occlusion-robust visual counting, statistical outlier detection, and symbolic cell state enumeration in cellular automata.

CountOCC designates a diverse set of methods, algorithms, and data structures devoted to counting structured objects in combinatorial, statistical, and computational frameworks. In contemporary literature, CountOCC appears in at least five major contexts: colored orthogonal range counting in computational geometry, enumeration of algebraic combinatorial sequences, pattern occurrence counting within permutations, robust amodal counting under vision domain occlusion, and meta-algorithms for counting cell states in cellular automata. The term "CountOCC" (Editor's term) will be used to refer to both canonical range-counting problems and their algorithmic generalizations.

1. Two-Dimensional Orthogonal Colored Range Counting

The 2D orthogonal colored range counting problem requires preprocessing a planar set $P$ of $n$ colored points so that, for any axis-aligned rectangle $R$, one can compute

$$\mathit{CountOCC}(R) = \left|\{c(p) : p \in P \cap R\}\right|$$

i.e., the number of distinct colors present in $R$ (Gao et al., 2021). Early work established reduction to higher-dimensional range searching, yielding quadratic space and polylogarithmic query time. The Kaplan–Rubin–Sharir–Verbin structure improved this to $O(n \lg^4 n)$ space and $O(\sqrt{n} \lg^8 n)$ time.

The state-of-the-art CountOCC framework comprises three tradeoff solutions, each leveraging binary range trees, reduction to canonical stabbing queries, and precomputed intersection matrices via block partitioning (parameter $X$):

Solution	Space Complexity	Query Complexity
Theorem 3.1	$O(n \lg^3 n)$	$O(\sqrt{n}\lg^{5/2} n \lg\lg n)$
Theorem 4.1	$O(n \lg^2 n)$	$O(\sqrt{n}\lg^{4+\epsilon} n)$
Theorem 4.2	$O(n\lg^2 n/\lg\lg n)$	$O(\sqrt{n}\lg^{5+\epsilon'} n)$
Linear-space	$O(n\lg n)$	$O(n^{1/2+\epsilon})$

These approaches utilize canonical box decompositions, efficient block matrix lookups for color intersection computations, and time–space tradeoffs parametrized by the block size $X$. A conditional lower bound, derived from Boolean matrix multiplication complexity, constrains purely combinatorial algorithms to $Q(n) = \Omega(\sqrt{n}/\text{polylog}(n))$ given near-linear space.

The new CountOCC solutions strictly improve polylogarithmic factors in both query time and space efficiency relative to prior combinatorial methods, approaching the optimal regime of $(n, \sqrt{n})$ (Gao et al., 2021).

2. Enumeration of Finite O-Sequences of Given Multiplicity

For Artinian graded algebras $R/I$, a finite $O$-sequence $H = (a_0, \ldots, a_s)$ is the Hilbert function with total multiplicity $d = \sum a_i$. The enumeration problem seeks

$O_d = \#\{\text{finite %%%%22%%%%-sequences of multiplicity }d\}$

(Cioffi et al., 31 Jul 2025).

The analysis decomposes $O_d$ via an “add–one–and–attach” process. Any $O$-sequence of multiplicity $d-1$ produces $O_{d-1}$ new $d$-sequences by appending $1$, while sequences with incrementable last entry yield $A_d$ further options: $O_d = O_{d-1} + A_d$ with $A_d$ bounded by

$$A_{d-2} \leq A_d \leq A_{d-1} + A_{d-2} = O_{d-1} - O_{d-3}$$

leading to the sub-Fibonacci inequality: $O_d \leq O_{d-1} + O_{d-2}$ for $d \geq 3$. The sequence $O_d$ thus grows strictly slower than Fibonacci, and the ratio $r_d = O_d/O_{d-1}$ remains bounded above by the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$ in the limit.

Iterative formulas, based on lex-segment decomposition, are given by: $$O(p, n, k, d) = \sum_{j=1}^{d-1} \sum_{i=k}^n O(p-1, n, i, d-j) \cdot O(p, i-1, k-1, j)$$ with

$O_d = O(d, d-1, 0, d)$

providing a complete recursive structure for practical enumeration. The direct enumeration algorithm leverages constructive steps subject to Macaulay’s condition and runs in $O(d)$, with published computations up to $d=60$.

Asymptotically, $O_d$ is bounded above by Fibonacci numbers, and the generating function $O(x)$ converges for $|x| < 1/\varphi$, satisfying $O(x) \leq x/(1-x-x^2)$ (Cioffi et al., 31 Jul 2025).

3. Counting Pattern Occurrences in Permutations (Permutation CountOCC)

Given a permutation $\pi \in S_n$ and a pattern $\sigma \in S_k$, the number of pattern occurrences is

$$\operatorname{occ}(\pi, \sigma) = \left|\{(i_1 < \cdots < i_k) : \text{std}(\pi_{i_1},\ldots,\pi_{i_k}) = \sigma\}\right|$$

(Conway et al., 2023). The distribution of occurrence counts $\psi_r(n; \sigma)$ can be extracted across all $\pi \in S_n$.

The multiset–BDD (Binary Decision Diagram) algorithm for CountOCC constructs a multiset $M$ of permutations in $S_n$ such that each appears with multiplicity equal to its occurrence count. The MZDD (multiset ZDD) represents this efficiently, allowing histogram extraction of $\psi_r(n;\sigma)$ via dynamic programming in $O(N M_{\max})$ time ($N$: node count; $M_{\max}$: maximal multiplicity).

For patterns of length 3, two Wilf classes yield closed formulas for $\psi_r(n;\sigma)$ and asymptotics, with the distribution approaching Gaussian scaling for $\sigma \in \{123,321\}$. Patterns of length 4 split into seven Wilf classes with distinct stretched-exponential and power-law behaviors in the zero-occurrence regime, governed by analytical series and conjectured closed forms. The MZDD method supports $n$ up to $\approx$14 in practice for $k=4$.

This unified approach enables fine-grained slicing of permutation space by pattern statistics, with open questions concerning algebraicity, D-finiteness, and multiset–BDD complexity (Conway et al., 2023).

4. Amodal Visual Counting Under Occlusion: Vision Domain CountOCC

CountOCC in vision refers to an amodal counting system designed to address feature corruption due to occlusion in object counting tasks (Arib et al., 16 Nov 2025). Standard backbone encoders misrepresent occluded regions, resulting in undercounts. CountOCC integrates:

• Feature Reconstruction Module (FRM): At each feature pyramid level, the FRM reconstructs occluded object features by fusing visible spatial context and semantic priors (text and visual embeddings) through staged self- and cross-attention layers. Occluded slots in the feature map are synthesized from learnable queries and replaced hierarchically.
• Visual Equivalence (VisEQ) Loss: Enforces consistency of gradient-based attention maps between occluded and unoccluded views using a teacher-student framework. Attention similarity ($\ell_2$ and cosine losses) and RoI consistency terms align spatial attention regardless of occlusion.

CountOCC employs a two-stage training process with occlusion augmentation and explicit distillation supervision. Evaluation on occlusion-augmented FSC-147, CARPK, and CAPTURe-Real benchmarks reports substantial relative reductions in mean absolute error: 26.72% and 20.80% (val and test, FSC-147), 49.89% (CARPK, zero-shot), and 28.79% (CAPTURe-Real).

Ablations confirm that hierarchical FRM yields the principal gains, while VisEQ sharpens outlier elimination and RMSE reduction. Loss component analysis demonstrates necessity of combined $\ell_2$, cosine, and Charbonnier terms. t-SNE projections verify tight clustering of reconstructed features with ground-truth, and successful cross-domain generalization (Arib et al., 16 Nov 2025).

5. Collective Outlier Detection and Enumeration: Statistical CountOCC

The CountOCC framework in outlier enumeration operates on a test set $Y_1,\ldots,Y_n$ with reference inliers $X_1,\ldots,X_m$. For every subset $S$ of indices, the intersection null hypothesis

$H_S := \bigwedge_{j \in S} (P_j = P_0)$

is tested using conformal rank-based p-values or permutation statistics (WMW, Fisher, Simes, LMPI). The closed testing principle adjusts local tests $\phi_S$ to compute simultaneously valid lower ($\ell$) and upper ($u$) bounds for the number of outliers, yielding a confidence set $\hat{\mathcal{M}}_\alpha$ for plausible outlier counts (Magnani et al., 2023).

Efficient computational shortcuts for closed testing are available with monotone symmetric tests (Simes: $O(n\log n)$, WMW/Fisher: $O(n)$), obviating combinatorial enumeration of $2^n$ subsets.

Finite-sample guarantees derive from exchangeability, yielding exact FWER control at level $\alpha$ and simultaneous lower bounds for all subsets. Asymptotics deploy CLT approximations for permutation tests.

In practical terms, the CountOCC outlier enumeration sets a statistically calibrated minimum count for true outliers—e.g., at $\alpha=0.1$, one may assert with 90% confidence that at least $d([n])$ points are outliers in a given test set. Extensions support upper bounds via complementary hypothesis testing (Magnani et al., 2023).

6. Meta-Algorithm for Counting ON Cells in Odd-Rule Cellular Automata

For cellular automata defined by a polynomial $P(x_1,\ldots,x_d)$ and prime $p$, CountOCC transforms the cell-counting problem at step $n$ into computation of $a(n) = [P(x)^n \bmod p]_{x=1}$ (Ekhad et al., 2015). The meta-algorithm comprises:

1. Construction of auxiliary sequences $a_Q(n)$ indexed by polynomials $Q(x)$.
2. Use of the “Freshman's dream” property in characteristic $p$ to derive finite $p$-ary recurrence relations with integer coefficient matrices $M_i$:

$a_j(pn + i) = \sum_{\ell} M_i[j,\ell] a_\ell(n)$

1. Recursive evaluation of $a(n)$ via base-$p$ digit decomposition and iterative matrix multiplication, yielding $O(\log n)$ run time for fixed $m$ (number of auxiliary polynomials).
2. The initial conditions and recurrence closure are computed by direct expansion for small $n$.

This method generalizes to arbitrary polynomials and primes, supports efficient implementation, and admits closed-form rational generating functions in the special case of subsequences $b(k) = a(p k - 1)$ (Ekhad et al., 2015).

7. Synthesis and Theoretical Implications

The CountOCC paradigm encompasses high-efficiency data structures for geometric queries, recursion-based combinatorial enumeration, advanced permutation statistics, state-of-the-art deep learning for occlusion-robust counting, rigorous statistical enumeration for robust inference, and meta-algorithms for symbolic cell-state counting. All instances share a focus on efficient, scalable counting of structured occurrences, be they colors, features, patterns, outliers, or polynomial coefficients, and leverage tree-based decompositions, dynamic programming, recurrences, or information-theoretic bounds as appropriate to the setting.

Recent advances have focused on improving space–time tradeoffs, sharpening asymptotic bounds, and ensuring robustness against adversarial or missing information (occlusion, outliers). Conditional lower bounds (e.g., via matrix multiplication) and sub-Fibonacci growth envelopes reveal natural computational and structural obstacles.

Open problems noted include algebraicity of generating functions for permutation statistics, theoretical bounds for decision diagram representations, and further explanation for stretched exponential regimes. The general CountOCC architecture underpins scalable solutions for modern counting problems in computational geometry, combinatorics, vision, statistics, and symbolic computation.