Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions

Baker, Andy; Males, Joshua

doi:10.1007/s00026-022-00612-4

Cited by 3 publications

(1 citation statement)

References 23 publications

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“…They conjectured that for any d ⩾ 1 the plane partition function fulfills the order d Turán inequality for all large enough numbers n. That conjecture was solved by Ono, Pujahari and Rolen in [41] with explicit bounds provided by Ono's PhD student Pandey [42]. Further, Baker and Males [8] showed that the number p j (n) of partitions with BG-rank j, and the number p j (a, b; n) of partitions with BG-rank j and 2-quotient rank congruent to a (mod b) satisfy (asymptotically) all higher order Turán inequalities for even values of j and n. We refer the reader to Berkovich and Garvan's paper [11] for additional information about p j (n) and p j (a, b; n). Finally, Dong, Ji and Jia [23] discovered that the Jensen polynomial corresponding to d ⩾ 1 and the Andrews and Paule's broken k-diamond partition function ∆ k (n), namely J d,n…”

Section: Introductionmentioning

confidence: 98%

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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Section: Introductionmentioning

confidence: 98%

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

Gajdzica

2023

Preprint

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Strict Log-Subadditivity for Overpartition Rank

Zhang

2023

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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Asymptotics, Turán Inequalities, and the Distribution of the BG-Rank and 2-Quotient Rank of Partitions

Cited by 3 publications

References 23 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

Strict Log-Subadditivity for Overpartition Rank

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

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