Parity distribution and divisibility of Mex-related partition functions Article Swipe

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Subhrajyoti Bhattacharyya , Rupam Barman , Ajit Singh , Apu Kumar Saha ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2303.03647

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(λ)$ for an integer partition $λ$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $λ$. They then defined $p_{A,a}(n)$ to be the number of partitions $λ$ of $n$ satisfying $\text{mex}_{A,a}(λ)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ and $t$. We show that for sufficiently large $X$, the number of all positive integer $n\leq X$ such that $p_{mt,t}(n)$ is an even number is at least $\mathcal{O}(\sqrt{X/3})$ for all positive integers $m$ and $t$. We also prove that for sufficiently large $X$, the number of all positive integer $n\leq X$ such that $p_{mp,p}(n)$ is an odd number is at least $\mathcal{O}(\log \log X)$ for all $m\not \equiv 0\pmod{3}$ and all primes $p\equiv 1\pmod{3}$. Finally, we establish identities connecting the ordinary partition function to $p_{mt,t}(n)$.

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Combinatorics Integer (computer science) Partition (number theory) Lambda Divisibility rule Mathematics Function (biology) Modulo Parity (physics) Physics Particle physics Optics Computer science Programming language Evolutionary biology Biology

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Type: preprint
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Landing Page: http://arxiv.org/abs/2303.03647
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DOI: https://doi.org/10.48550/arxiv.2303.03647

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Title: Parity distribution and divisibility of Mex-related partition functions

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Publication year: 2023

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Authors: Subhrajyoti Bhattacharyya, Rupam Barman, Ajit Singh, Apu Kumar Saha

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Concepts: Combinatorics, Integer (computer science), Partition (number theory), Lambda, Divisibility rule, Mathematics, Function (biology), Modulo, Parity (physics), Physics, Particle physics, Optics, Computer science, Programming language, Evolutionary biology, Biology

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