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ON NEW MINIMAL EXCLUDANTS OF OVERPARTITIONS RELATED TO SOME q-SERIES OF RAMANUJAN
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 11 February 2025, pp. 1-13
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ON A CONJECTURE FOR A REFINEMENT OF THE SUM OF MINIMAL EXCLUDANTS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 20 July 2023, pp. 276-287
- Print publication:
- April 2024
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In 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23(2) (2019), 249–254] introduced the arithmetic function
$\sigma \textrm {mex}(n)$, which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo 
$4$ and 
$8$ for two functions appearing in this refinement and conjectured two further congruences modulo 
$8$ and 
$16$. We confirm these two conjectures by using q-series manipulations and modular forms.
