Rogers-Ramanujan Type Identities for Burge’s Restricted Partition Pairs Via Restricted Frobenius Partitions

Agarwal, Ashok; Padmavathamma,

doi:10.1007/978-3-0348-8223-1_6

Cited by 2 publications

(5 citation statements)

References 3 publications

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“…p(n) = #P (n). For instance, we have that p(4) = 5 with (4), (3,1), (2,2), (2, 1, 1) and (1, 1, 1, 1).…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, there is a plethora of literature devoted to its analytic, arithmetic and combinatorial properties. We refer the reader to [1,5,6,20,32] for a comprehensive introduction to the theory of partitions. Now, let us recall that a sequence (c n ) ∞ n=0 of real numbers is said to be logconcave if it fulfills c 2 n > c n−1 c n+1 for every n ⩾ 1.…”

Section: Introductionmentioning

confidence: 99%

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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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“…p(n) = #P (n). For instance, we have that p(4) = 5 with (4), (3,1), (2,2), (2, 1, 1) and (1, 1, 1, 1).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gajdzica

2024

Preprint

View full text Add to dashboard Cite

show abstract

“…The numbers λ i are called parts of the partition λ. The partition function p(n) enumerates all partitions of n. For instance, there are 5 partitions of 4, namely, (4), (3,1), (2,2), (2, 1, 1) and (1, 1, 1, 1) -in other words p(4) = 5. We do not know any easy formula for p(n).…”

Section: Introductionmentioning

confidence: 99%

“…There is a plethora of works devoted to the theory of partitions. For a general introduction to the topic, we encourage the reader to see Andrews' books [4,5] as well as [1,31,45]. Now, let us assume that A = {a 1 , a 2 , .…”

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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Rogers-Ramanujan Type Identities for Burge’s Restricted Partition Pairs Via Restricted Frobenius Partitions

Cited by 2 publications

References 3 publications

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

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

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

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