On Conjectures Concerning the Smallest Part and Missing Parts of Integer Partitions

Binner, Damanvir Singh; Rattan, Amarpreet

doi:10.1007/s00026-021-00528-5

Cited by 2 publications

(5 citation statements)

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“…num. of [BR21] not in range of φ of partitions 1, in Rows 1-4 a straight forward application of Proposition 7 to n = N − ts, where t is the number of parts of s, and a and b are set to be the remaining parts, which are coprime, gives the numbers in Column 4.…”

Section: Case Of L Smentioning

confidence: 99%

“…As indicated by Theorem 1, the proof in[BR21] for the positivity of H L,s,k (q) applies for all k 󰃍 s + 1. In particular, a more general I L,s,k is defined than the one given in (14).…”

mentioning

confidence: 91%

“…Proof. The proof of Theorem 1 in [BR21] uses the combinatorial interpretation of the coefficients of H L,s,L (q) in (14); the proof there constructs for N 󰃍 Γ(s) an injection as in (15) to show nonnegativity of (14). To show positivity of (14), elements of the codomain that are not in the range of φ are given.…”

mentioning

confidence: 99%

“…To show positivity of (14), elements of the codomain that are not in the range of φ are given. The details of these injections and elements are dealt with in different cases and theorems of [BR21] depending on the relative sizes of L and s. 1 For example, when L 󰃍 2s + 3, there is no partition of the form (s 10 , (s + 1) x , (s + 2) y ) in the range of φ, but such partitions are in the codomain. To achieve the present result, we count the number of such partitions; this will then give the desired lower bound for the difference (14) in each case.…”

mentioning

confidence: 99%

“…We then compare the results of all cases. Table 1 lists the cases, the theorem from [BR21] showing positivity of ( 14), the partitions in the codomain but not the range of the relevant injection φ in (15), and a lower bound for the enumeration of these partitions. To count the number of partitions in Table 1: The first column describes the case, the second the theorem from [BR21] that proves positivity in this case, the third the partitions in the codomain that are not in the range of φ in (15), and the last column contains the minimum number of partitions of the type in Column 3 .…”

mentioning

confidence: 99%

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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: Case Of L Smentioning

confidence: 99%

“…As indicated by Theorem 1, the proof in[BR21] for the positivity of H L,s,k (q) applies for all k 󰃍 s + 1. In particular, a more general I L,s,k is defined than the one given in (14).…”

mentioning

confidence: 91%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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