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Further results on biases in integer partitions
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Let $p_{a,b,m}(n)$ be the number of integer partitions of $n$ with more parts congruent to $a$ modulo $m$ than parts congruent to $b$ modulo $m$. We prove that $p_{a,b,m}(n)\ge p_{b,a,m}(n)$ whenever $1\le a<b\le m$. We also propose some conjectures concerning series with nonnegative coefficients in their expansions.
Abstract
Let $p_{a,b,m}(n)$ be the number of integer partitions of $n$ with more parts congruent to $a$ modulo $m$ than parts congruent to $b$ modulo $m$. We prove that $p_{a,b,m}(n)\ge p_{b,a,m}(n)$ whenever $1\le a<b\le m$. We also propose some conjectures concerning series with nonnegative coefficients in their expansions.
참고문헌 (5)
- G. E. Andrews, "On a partition problem of H. L. Alder", Pacific J. Math, 36, pp.279~284 (1971)
- G. E. Andrews, "The theory of partitions, reprint of the 1976 original, Cambridge Mathematical Library" (1976), Cambridge University Press
- P. L. Chebyshev, "Lettre de M. le Professeur Tch´ebychev a M. Fuss sur un nouveaux theoreme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3", Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg, 11 (1853)
- Shane Chern, "Further results on biases in integer partitions", 대한수학회보, 59(1), pp.111~117 (2022), 대한수학회[DOI]
- Byungchan Kim, "Parity bias in partitions", European Journal of Combinatorics, 89, pp.103159~ (2020), Elsevier BV[DOI]
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