Analytic Combinatorics

By Philippe Flajolet, Robert Sedgewick. 2009

3,964 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/d347dbb4b2eea7fca0183b55112b9cc07faa51ff

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Citations in the corpus

A Tight Convergence Analysis for Stochastic Gradient Descent with Delayed Updates (2018, background): “We now turn to brieﬂy describe these functions and our approach (for general surveys on generating functions, see [25, 9, 23], to name a few).”
Scaling Limits of Random Graphs from Subcritical Classes (2014, methodology): “The reader is referred to the excellent sources Joyal [20], Bergeron, Labelle and Leroux [6] and Flajolet and Sedgewick [15] for an introduction to the framework of combinatorial classes and generating functions.”
Mod-ϕ Convergence (2016, background): “There exist several other versions of this theorem where one can replace log(1− t/r) by other functions and we refer the reader to the monograph [FS09], chapter VI.”
The Weibull Marshall–Olkin family: Regression model and application to censored data (2018, background): “The proof is given in details by Flajolet and Odlyzko (1990) (see Theorem 3A, page 227) and Flajolet and Sedgewick (2009) (see Theorem VI.2, page 385).”
Analytic Combinatorics in Several Variables (2013, background): “Its chief objects of study are generating functions (Flajolet and Sedgewick, 2009, page vii).”
A Survey of Alternating Permutations (2009, methodology): “ En n! = 2 2 π n+1 X k≥0 (−1)k(n+1) 1 (2k+1)n+1 . This formula forP nodd is equivalent to the well-known evaluation of ζ(n+ 1) := k≥1 k −(n+1). For further information on asymptotic expansions see [23], where in particular Euler numbers are treated in Example IV.35. 2. Reﬁnements of Euler numbers F”
Universal limits of substitution-closed permutation classes (2017, background): “The complete statement of Daffodil Lemma in [26] also deals with cases where the function A is periodic, but we do not need these cases in our work.”
Probability on Trees and Networks (2017, background): “6 and elsewhere in Flajolet and Sedgewick (2009). 6.61. (This is due to Lyons and Peres (2015a).) For (c), consider concatenating a long cycle that starts with e and ends with −e with a short cycle that does not.”
Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation (2011, neutral): “lytic at ρZ, Z(z) = f1(z) + a1(ρZ −z)α + o |ρZ −z|α , as z → ρZ, (17.36) where a1 = a ρΦ(ρ)α−1 (1 −ν)α+1 . (17.37) As noted above, Z(z) has no other singularities on |z| = ρ, and singularity analysis [40] applies and shows, using (17.33), Zn ∼ a1 Γ(−α) n−α−1ρα−n Z ∼ ρ (1 −ν)α+1 Φ(ρ)n−1w n. (17.38) H”