A note on the restricted partition function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">A</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

Gajdzica, Krystian

doi:10.1016/j.disc.2022.112943

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…There is an abundance of literature devoted to A-partition function when #A < ∞. We refer the reader to, for instance, [2,10,15,21,25,37,44,49].…”

Section: Introductionmentioning

confidence: 99%

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Restricted partition functions and the r-log-concavity of quasi-polynomial-like functions

Gajdzica

2023

Preprint

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Let A = (ai)∞ i=1 be a weakly increasing sequence of positive integers and let k be a fixed positive integer. For an arbitrary integer n, the restricted partition pA(n, k) enumerates all the partitions of n whose parts belong to the multiset {a1, a2, . . . , ak}. In this paper we investigate some generalizations of the log-concavity of pA(n, k). We deal with both some basic extensions like, for instance, the strong log-concavity and a more intriguing challenge that is the r-log-concavity of both quasi-polynomial-like functions in general, and the restricted partition function in particular. For each of the problems, we present an efficient solution.

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“…There is an abundance of literature devoted to A-partition function when #A < ∞. We refer the reader to, for instance, [2,10,15,21,25,37,44,49].…”

Section: Introductionmentioning

confidence: 99%

“…There is an abundance of literature devoted to the restricted partition function. For arithmetic properties of p A (n, k), we refer the reader to [15,31,38]. On the other hand, for some information about the asymptotic behavior we encourage to see [1,8,12].…”

Section: Introductionmentioning

confidence: 99%

Restricted partition functions and the r-log-concavity of quasi-polynomial-like functions

Gajdzica

2023

Preprint

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Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions

Gajdzica,

Heim,

Neuhauser

2024

Ann. Comb.

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The Turán and Laguerre inequalities for quasi-polynomial-like functions

Gajdzica

2024

Res. number theory

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This paper deals with both the higher order Turán inequalities and the Laguerre inequalities for quasi-polynomial-like functions that are expressions of the form $$f(n)=c_l(n)n^l+\cdots +c_d(n)n^d+o(n^d)$$ f ( n ) = c l ( n ) n l + ⋯ + c d ( n ) n d + o ( n d ) , where $$d,l\in \mathbb {N}$$ d , l ∈ N and $$d\leqslant l$$ d ⩽ l . A natural example of such a function is the A-partition function $$p_{A}(n)$$ p A ( n ) , which enumerates the number of partitions of n with parts in the fixed finite multiset $$A=\{a_1,a_2,\ldots ,a_k\}$$ A = { a 1 , a 2 , … , a k } of positive integers. For an arbitrary positive integer d, we present efficient criteria for both the order d Turán inequality and the dth Laguarre inequality for quasi-polynomial-like functions. In particular, we apply these results to deduce non-trivial analogues for $$p_A(n)$$ p A ( n ) .

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A note on the restricted partition function pA(n,k)

Cited by 3 publications

References 17 publications

Restricted partition functions and the r-log-concavity of quasi-polynomial-like functions

Restricted partition functions and the r-log-concavity of quasi-polynomial-like functions

Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions

The Turán and Laguerre inequalities for quasi-polynomial-like functions

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