Partitions With Parts in a Finite Set

Rødseth, Øystein J.; Sellers, James A.

doi:10.1142/s1793042106000644

Cited by 10 publications

(6 citation statements)

References 14 publications

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“…Arithmetic properties of p A (n, k) were explored especially by Nathanson [14], Almkvist [1] and Rødseth and Sellers [20]. In this paper we mainly investigate periodicity and both upper and lower bounds for odd density of the function p A (n, k) for a fixed positive integer k. In particular, we generalize results obtained in [11] by Karhadkar.…”

Section: Introductionmentioning

confidence: 86%

A note on the restricted partition function $p_\mathcal{A}(n,k)$

Gajdzica

2021

Preprint

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Let A = (an) n∈N + be a sequence of positive integers. Let p A (n, k) denote the number of multi-color partitions of n into parts in {a 1 , . . . , a k }. We examine several arithmetic properties of the sequence (p A (n, k) (mod m)) n∈N for an arbitrary fixed integer m 2. We investigate periodicity of the sequence and lower and upper bounds for the density of the set {n ∈ N : p A (n, k) ≡ i (mod m)} for a fixed positive integer k and i ∈ {0, 1, . . . , m−1}. In particular, we apply our results to the special cases of the sequence A. Furthermore, we present some results related to restricted m-ary partitions.

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Section: Introductionmentioning

confidence: 86%

A note on the restricted partition function $p_\mathcal{A}(n,k)$

Gajdzica

2021

Preprint

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“…The notion of quasi polynomial seems to be subsist from the time of Bell [1], who proved that the partition function, p A (n)-number of partitions of n with parts from a finite set of positive integers A, is a quasi polynomial with each constituent polynomial being of degree at most |A| − 1 and quasi period being a positive common multiple of elements of A. This fact was also proved recently by Rødseth and Sellers [5].…”

Section: Introductionmentioning

confidence: 87%

Estimates of five restricted partition functions that are quasi polynomials

Christopher

Christober

2014

Bull. Math. Sci.

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A function f defined on N is said to be a quasi polynomial if, f (αn + r) is a polynomial in n for each r = 0, 1,. .. , α − 1, where α is a positive integer. In this article, we show that the below given restricted partition functions are quasi polynomials: (i) a(n, k)-number of partitions of n with exactly k parts and least part being less than k, (ii) aq(n, k)-number of distinct partitions (partitions with distinct parts) of n with exactly k parts and least part being less than k, (iii) Le(n, k, m)-number of partitions of n with exactly k parts and m least parts, (iv) La(n, k, 1)-number of partitions of n with exactly k parts and one largest part and (v) d(n, k)-number of partitions of n with exactly k parts and difference between least part and largest part exceeds k − 2. Consequently, following estimates were derived: (i) a(n, k) ∼ n k−2 (k − 2)! 2 (ii) aq(n, k) ∼ n k−2 (k − 2)! 2 Communicated by Ari Laptev.

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

confidence: 98%

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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Partitions With Parts in a Finite Set

Cited by 10 publications

References 14 publications

A note on the restricted partition function $p_\mathcal{A}(n,k)$

A note on the restricted partition function $p_\mathcal{A}(n,k)$

Estimates of five restricted partition functions that are quasi polynomials

Some general results in combinatorial enumeration

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