Abstract
We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for Lge 1, the number of partitions with l-s le L and s=1 is greater than the number of partitions with l-sle L and s>1. Here l and s are the largest part and the smallest part of the partition, respectively.
Highlights
Let π = (1f1, 2f2, . . . ) be a sequence, where all exponents fi ∈ Z≥0 and all but finitely many of them are zero
We introduce that result and its refinement here
In [10], we introduced a new partition statistic t(π) to be the number defined by the properties (i) fi ≡ 1 mod 2, for 1 ≤ i ≤ t(π), (ii) ft(π)+1 ≡ 0 mod 2
Summary
Elementary combinatorial inequalities, such as (1.1), have interesting implications for q-series and the theory of partitions This simple observation about the magnitude of sets, in this case, implies non-negativity results for a refinement of an earlier discussed-weighted partition identity result [10]. We will refer to t(π) as the length of the initial odd-frequency chain With this new statistic, the authors have proven a new combinatorial identity of partitions. It is clear that t(π)q|π| = pt(n)qn 0, π∈U n≥1 where pt(n) is the total weighted count of partitions with the t statistic This implies that the series in (1.2) is non-negative. We introduce a refinement of Theorem 1.2 where we put a bound on the difference between the largest and the smallest parts of partitions. An outlook section finishes the paper with a summary of open questions that arise from this study
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