We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and M... more We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and Miska (Results Math, 2021. https://doi.org/10.1007/s00025-021-01479-2) and demonstrate that the original construction of a special series with unique subsums remains valid when using a weaker estimate that we prove to be true. Additionally, we present a weaker version—without the uniqueness of subsums—of the Thm. 2.1 from Marchwicki and Miska (2021), but with a very simple proof based on the concept of semi-fast convergent series.
We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and M... more We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and Miska (Results Math, 2021. https://doi.org/10.1007/s00025-021-01479-2) and demonstrate that the original construction of a special series with unique subsums remains valid when using a weaker estimate that we prove to be true. Additionally, we present a weaker version—without the uniqueness of subsums—of the Thm. 2.1 from Marchwicki and Miska (2021), but with a very simple proof based on the concept of semi-fast convergent series.
Let $$0\le q\le 1$$ 0 ≤ q ≤ 1 and $$\mathbb {N}$$ N denotes the set of all positive integers. In ... more Let $$0\le q\le 1$$ 0 ≤ q ≤ 1 and $$\mathbb {N}$$ N denotes the set of all positive integers. In this paper we will deal with it too the family $${\mathcal {U}}(x^q)$$ U ( x q ) of all regularly distributed set $$X=\{x_1
The classical derangement numbers count fixed point-free permutations. In this paper we study the... more The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, p-adic valuations, and diophantine problems.
We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and M... more We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and Miska (Results Math, 2021. https://doi.org/10.1007/s00025-021-01479-2) and demonstrate that the original construction of a special series with unique subsums remains valid when using a weaker estimate that we prove to be true. Additionally, we present a weaker version—without the uniqueness of subsums—of the Thm. 2.1 from Marchwicki and Miska (2021), but with a very simple proof based on the concept of semi-fast convergent series.
We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and M... more We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and Miska (Results Math, 2021. https://doi.org/10.1007/s00025-021-01479-2) and demonstrate that the original construction of a special series with unique subsums remains valid when using a weaker estimate that we prove to be true. Additionally, we present a weaker version—without the uniqueness of subsums—of the Thm. 2.1 from Marchwicki and Miska (2021), but with a very simple proof based on the concept of semi-fast convergent series.
Let $$0\le q\le 1$$ 0 ≤ q ≤ 1 and $$\mathbb {N}$$ N denotes the set of all positive integers. In ... more Let $$0\le q\le 1$$ 0 ≤ q ≤ 1 and $$\mathbb {N}$$ N denotes the set of all positive integers. In this paper we will deal with it too the family $${\mathcal {U}}(x^q)$$ U ( x q ) of all regularly distributed set $$X=\{x_1
The classical derangement numbers count fixed point-free permutations. In this paper we study the... more The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, p-adic valuations, and diophantine problems.
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