Proof of the Bessenrodt–Ono Inequality by Induction

Heim, Bernhard; Neuhäuser, Markus

doi:10.1007/s40993-021-00301-w

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…Recently, we invented a new proof method [18] and reproved some known results related to the Bessenrodt-Ono inequality for the partition function, the k-colored partitions and extension to the D'Arcais polynomials.…”

Section: Related Work and Recent Resultsmentioning

confidence: 99%

Inequalities for Plane Partitions

2022

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Section: Related Work and Recent Resultsmentioning

confidence: 99%

Inequalities for Plane Partitions

2022

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“…The first of them is in combinatorial manner due to Alanazi, Gagola and Munagi [2]. The second one, on the other hand, is done via induction on a + b by Heim and Neuhauser [17].…”

Section: Introductionmentioning

confidence: 99%

On the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions

Gajdzica

2024

Preprint

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The $A$-partition function $p_A(n)$ enumerates those partitions of $n$ whose parts belong to a fixed (finite or infinite) set $A$ of positive integers. On the other hand, the extended $A$-partition function $p_A\left(\bm{\mu}\right)$ is defined as an multiplicative extension of the $A$-partition function to a function on $A$-partitions. In this paper, we investigate the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions. In particular, we examine the property for both the $m$-ary partition function $b_m(n)$ and the $d$-th power partition function $p_d(n)$. Moreover, we show that $b_m(\bm{\mu})$ ($p_d(\bm{\mu})$) takes its maximum value at an explicitly described set of $m$-ary partitions (power partitions), where $\bm{\mu}$ is an $m$-ary partition (a power partition) of $n$. Additionally, we exhibit analogous results for the Fibonacci partition function and the 'factorial' partition function. It is worth pointing out that an elementary combinatorial reasoning plays a crucial role in our investigation.

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“…Alanazi, Gagola and Munagi [1] showed the Bessenrodt-Ono inequality in combinatorial manner by constructing appropriate injections between some sets of partitions. Heim and Neuhauser [19], on the other hand, presented a proof which is based on the induction on a + b.…”

Section: Introductionmentioning

confidence: 99%

On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties

Gajdzica,

Miska,

Ulas

2024

Ann. Comb.

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In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function F of at most exponential growth satisfying the condition $$F(\mathbb {N})\subset \mathbb {R}_{+}$$ F ( N ) ⊂ R + , we have $$F(a)F(b)>F(a+b)$$ F ( a ) F ( b ) > F ( a + b ) for sufficiently large positive integers a, b. Moreover, we show that if the sequence $$(F(n))_{n\ge n_{0}}$$ ( F ( n ) ) n ≥ n 0 is log-concave and $$\limsup _{n\rightarrow +\infty }F(n+n_{0})/F(n)<F(n_{0})$$ lim sup n → + ∞ F ( n + n 0 ) / F ( n ) < F ( n 0 ) , then F satisfies the Bessenrodt–Ono type inequality.

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Proof of the Bessenrodt–Ono Inequality by Induction

Cited by 5 publications

References 24 publications

Inequalities for Plane Partitions

Inequalities for Plane Partitions

On the Bessenrodt-Ono type inequality for a wide class of $A$-partition functions

On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties

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