Some Elementary Partition Inequalities and Their Implications

Berkovich, Alexander; Uncu, Ali Kemal

doi:10.1007/s00026-019-00433-y

Cited by 29 publications

(4 citation statements)

References 11 publications

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“…For fixed L and s, let p L,s (q) be the smallest degree polynomial in q with smallest coefficients such that G L,s (q) + p L,s (q) ≽ 0. This paper finds p L,s (q) for s = 3 and all L, while the cases s = 2 (found in Theorem 2) and s = 1 (found in [BU19]) were found earlier. One goal is to find p L,s (q) for general s and L. Our numerical computations for s = 4 and s = 5 do not suggest a nice form for p L,s (q).…”

Section: 󰀔supporting

confidence: 47%

A Comparison of Integer Partitions Based on Smallest Part

Binner,

Amarpreet Rattan

2024

Electron. J. Combin.

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For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu (2019). Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general $s$ and also find a precise nonnegativity result for the case $s=3$.

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Section: 󰀔supporting

confidence: 47%

A Comparison of Integer Partitions Based on Smallest Part

Binner,

Amarpreet Rattan

2024

Electron. J. Combin.

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“…Moreover, the series F 1 (q) plays an instrumental role in Euler's original proof of Theorem 1.1 [1]. Recently F i,M (q) for i = 1, 2, and 3 arose naturally in our studies of partitions with bounded gaps between largest and smallest parts [3]. In the following sections, among other observations, we will prove the next two theorems.…”

Section: Introductionmentioning

confidence: 83%

On some polynomials and series of Bloch–Pólya type

Berkovich

Uncu

2018

Proc. Amer. Math. Soc.

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We will show that (1 − q)(1 − q 2 ) . . . (1 − q m ) is a polynomial in q with coefficients from {−1, 0, 1} iff m = 1, 2, 3, or 5 and explore some interesting consequences of this result. We find explicit formulas for the q-series coefficients of (1 − q 2 )(1 − q 3 )(1 − q 4 )(1 − q 5 ) . . . and (1 − q 3 )(1 − q 4 )(1 − q 5 )(1 − q 6 ) . . . . In doing so, we extend certain observations made by Sudler in 1964. We also discuss the classification of the products (1 − q)(1 − q 2 ) . . . (1 − q m ) and some related series with respect to their absolute largest coefficients.

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“…Besides, Kim, Kim and Lovejoy [10] recently investigated overpartition differences and their weighted generating functions and deduced a variety of positivity results for such differences. For other recent results on positivity for partition functions, we refer to [5,6].…”

Section: El Bachraouimentioning

confidence: 99%