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In I981, Uchimura studied a divisor generating $q$-series that has applications in probability theory and in the analysis of data structures, called heaps. Mainly, he proved the following identity. For $|q|<1$, \begin{equation*} \sum_{n=1}^\infty n q
Externí odkaz:
http://arxiv.org/abs/2405.01877
The minimal excludant (mex) of a partition was introduced by Grabner and Knopfmacher under the name `least gap' and was revived by a couple of papers due to Andrews and Newman. It has been widely studied in recent years together with the complementar
Externí odkaz:
http://arxiv.org/abs/2312.02620
In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity
Externí odkaz:
http://arxiv.org/abs/2210.03457
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a positive int
Externí odkaz:
http://arxiv.org/abs/2208.03658
Publikováno v:
In Discrete Mathematics December 2024 347(12)
We find an interesting refinement of a result due to Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number $n$ equals the number of distinct parts partitions of $n$ into two colors. In addition, we also study
Externí odkaz:
http://arxiv.org/abs/2110.08108
Publikováno v:
Int. J. Number Theory, 2022
The average size of the "smallest gap" of a partition was studied by Grabner and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work of Fraenkel and Peled, studied the concept of the "smallest gap" under the name "minimal excluda
Externí odkaz:
http://arxiv.org/abs/2105.13875
Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities
Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the thi
Externí odkaz:
http://arxiv.org/abs/2011.07767
Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $\varphi$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving u
Externí odkaz:
http://arxiv.org/abs/2001.05944