On the restricted partition function

Cimpoeaş, Mircea; Nicolae, Florin

doi:10.1007/s11139-017-9985-3

Cited by 20 publications

(20 citation statements)

References 14 publications

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“…, a k ) is a sequence of positive integers and p a (n) is the restricted partition function associated to a, we denote P a (n), the polynomial part of p a (n). Several formulas of P a (n) were proved in [2], [7] and [4].…”

Section: Resultsmentioning

confidence: 99%

A note on the number of partitions of $n$ into $k$ parts

Cimpoeaş¹

2021

Preprint

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Section: Resultsmentioning

confidence: 99%

A note on the number of partitions of $n$ into $k$ parts

Cimpoeaş¹

2021

Preprint

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“…A. Sellers [128], M. Cimpoeaş and F. Nicolae [41,42], the inductive proof of R. Jakimczuk [78] or the recent S. Robins and Ch. Vignat [127].…”

Section: Steps ✷mentioning

confidence: 98%

“…P k (n) are the partitions of n with k parts, and similarly Q k (n) are those with k distinct parts. The sequence p : N → N is[154, A000041] and begins (p(n)) n≥1 =(1,2,3,5,7,11,15,22,30,42,56,77,101,135, 176, . .…”

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confidence: 99%

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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“…In [6], we proved that the computation of p a (n) can be reduced to solving the linear congruency a 1 j 1 + • • • + a r j r ≡ n (mod D) in the range 0 ≤ j 1 ≤ D a1 , . .…”

Section: Introductionmentioning

confidence: 99%

On the restricted partition function

Cited by 20 publications

References 14 publications

A note on the number of partitions of $n$ into $k$ parts

A note on the number of partitions of $n$ into $k$ parts

Some general results in combinatorial enumeration

On the restricted partition function via determinants with Bernoulli polynomials. II

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