Convolutions of Liouvillian Sequences

Абрамов, С. А.; Petkovšek, Marko; Zakrajšek, Helena

doi:10.48550/arxiv.1803.08747

Cited by 1 publication

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“…The exponent 30 in the proof of Proposition 3.2 is hilarious but the reader understands that we do not optimize bounds and instead focus on simplicity of arguments. We can decrease it by computing the product a(x)b(x) more quickly but in a less elementary way in O((mn) 1+o (1) ) steps by [ 1) ) steps, again in close to optimum complexity. F. Johansson reports computing p(10 6 ) by his algorithm in milliseconds and p (10 19 ) in less than 100 hours; see [81] for his computation of p(10 20 ).…”

Section: Integer Partitionsmentioning

confidence: 99%

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Some general results in combinatorial enumeration

Klazar

2010

Permutation Patterns

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For enumerative problems, i.e. computable functions f : N → Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f (n) in the number of steps that is polynomial in the combined size of the input n and the output f (n), both written in binary notation. We discuss many examples of enumerative problems for which such closed formulas are, or are not, known. These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we discuss problems (i) and (ii) and defer (iii)-(v) to part II. Besides other results, we prove here that every linear recurrence sequence of integers has an effective formula in our sense. Definition 1.1 (PIO formula). For a counting function f : N → Z, a PIO formula is an algorithm, called a PIO algorithm, that for some constants c, d ∈ N for every input n ∈ N computes the output f (n) ∈ Z in at most c · m(n) d = O(m(n) d ) = poly(m(n)) steps, where m(n) = m f (n) := log(1 + n) + log(2 + |f (n)|) measures the combined complexity of the input and the output. Similarly for counting function f : X → Z defined on a subset X ⊂ N.We think this is the precise and definitive notion of a "closed formula", and the yardstick one should use, possibly with some ramifications or weakenings, to determine if a solution to an enumerative problem is effective. We call counting functions possessing PIO formulas shortly PIO functions. Definition 1.1 repeats

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Section: Integer Partitionsmentioning

confidence: 99%

“…A. Fill, S. Janson and M. D. Ward [56] proved that f dm (n) = exp((1 + o(1)) 1 3 (6n) 1/3 log n) and D. Kane and R. C. Rhoades [84] obtained an even more precise asymptotics.…”

Section: Computation-wise For P(n) It Lags Far Behind the H-r-r Formu...mentioning

confidence: 99%

Some general results in combinatorial enumeration

Klazar

2010

Permutation Patterns

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Convolutions of Liouvillian Sequences

Cited by 1 publication

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Some general results in combinatorial enumeration

Some general results in combinatorial enumeration

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