The paper provides the conjectured solution for the generating function of k-convex polyominoes, based on analysis of generated enumeration data.

This paper derives multivariate generating functions to refine the enumeration of Fibonacci polyominoes.

This paper introduces a novel algorithm for generating k Hamming weight binary words in linear time while minimizing random bit consumption.

The paper proposes a novel set of combined cellular automaton (CA)-based pseudo-random number generators (PRNGs) that overcome the weak equidistribution issues of existing CA-based PRNGs, achieving ma…

The paper analyzes language generation and identification in the limit under bounded memory, showing that memory constraints significantly alter learnability, particularly affecting achievable density…

This paper measures the lower bound for the shortest program generating a sequence, proving a conservation law and providing a deterministic engine to recover generating programs for certain sequences…

The paper proves that the reversible elementary second order cellular automaton rule 115 is periodic when started on finite initial configurations.

This book provides a compact, derivation-oriented mathematical primer that connects major families of generative AI models, showing their underlying structural relationships.

This paper analyzes the computational complexity of evaluating recurrent functions, showing that the complexity depends heavily on how the input offsets are encoded and the structure of the recurrence…

This paper traces the historical development of the square-and-multiply algorithm, arguing that while its conceptual foundation dates back to ancient Indian prosody, its formal articulation as a gener…

This paper systematically analyzes binomial functions over $\mathbb{F}_{p^n}$ in characteristic 3, providing a classification and rigorous proof of specific classes of exponents that yield extremely l…

This paper presents methods for ranking and unranking permutations avoiding a pattern of length three in lexicographic or colexicographic order.

This paper provides a probabilistic derivation of multiparameter signless Stirling numbers and their q-analogues.

The paper introduces the Insertion Process (IP), a novel stochastic generative model that learns variable-length, non-monotonic sequence generation by explicitly modeling the insertion order of tokens…

This paper presents an algorithmic framework for exhaustively generating and tabulating knot and link diagrams on the thickened torus.

This paper characterizes the graph structure, including cycle and path lengths, of Chebyshev permutation polynomials over the ring $\mathbb{Z}_{2^{k_1}3^{k_2}}$, demonstrating strong regularities desp…

This paper presents an algorithm to generate snake-like polyominoes within a given rectangle and provides exact formulas for the maximal area of such polyominoes for certain dimensions.

The paper introduces Conformal Generation (Conf-Gen), a novel framework that adapts conformal risk control to provide formal uncertainty guarantees for unsupervised generative models like LLMs and ima…

The paper establishes a strong connection between scalable pseudorandom unitaries (PRUs) and the unitary synthesis problem, proving that any such PRU construction must require a classical oracle of si…