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The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics…
The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate…
Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several…
Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that…
Rodrigo de Miguel et al 2007 J. Phys. A: Math. Theor. 40 5241-5260: A noisy vector channel operating under a strict complexity constraint at the receiver is introduced. According to this constraint, detected bits, obtained by performing…
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
The field of analytic combinatorics in several variables (ACSV) develops techniques to compute the asymptotic behaviour of multivariate sequences from analytic properties of their generating functions. When the generating function under…
Analytic combinatorics in several variables refers to a suite of tools that provide sharp asymptotic estimates for certain combinatorial quantities. In this paper, we apply these tools to determine the Gilbert-Varshamov (GV) bound for the…
Constrained multi-agent reinforcement learning offers the framework to design scalable and almost surely feasible solutions for teams of agents operating in dynamic environments to carry out conflicting tasks. We address the challenges of…
We introduce a resource theory of channels relevant to communication via quantum channels, in which the set of constant channels --- useless channels for communication tasks --- is considered as the free resource. We find that our theory…
A joint communication and channel state estimation problem is investigated, in which reliable information transmission over a noisy channel, and high-fidelity estimation of the channel state, are simultaneously sought. The tradeoff between…
In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko…
We introduce combinatorial interpretability, a methodology for understanding neural computation by analyzing the combinatorial structures in the sign-based categorization of a network's weights and biases. We demonstrate its power through…
Arbitrary varying channels (AVC) are used to model communication settings in which a channel state may vary arbitrarily over time. Their primary objective is to circumvent statistical assumptions on channel variation. Traditional studies on…
In this paper, we study an under-explored but important factor of diffusion generative models, i.e., the combinatorial complexity. Data samples are generally high-dimensional, and for various structured generation tasks, additional…
Arbitrarily varying channels offer a powerful framework for analyzing the robustness of quantum communication systems, especially for classical-quantum models, where the analysis displays strengths or weaknesses of specific signal…
When transmitting information over a noisy channel, two approaches, dating back to Shannon's work, are common: assuming the channel errors are independent of the transmitted content and devising an error-correcting code, or assuming the…
Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate…
In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…