Abstract
We investigate properties of attainable partitions of integers, where a partition \((n_1,n_2, \dots , n_r)\) of n is attainable if \(\sum (3-2i)n_i\ge 0\). Conjecturally, under an extension of the Cohen and Lenstra heuristics by Holmin et. al., these partitions correspond to abelian p-groups that appear as class groups of imaginary quadratic number fields for infinitely many odd primes p. We demonstrate a connection to partitions of integers into triangular numbers, construct a generating function for attainable partitions, and determine the maximal length of attainable partitions.
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Holmin, S., Jones, N., Kurlberg, P., McLeman, C., Petersen, K.: Missing class groups and class number statistics for imaginary quadratic fields. Exp. Math. 28(2), 233–254 (2019)
Cohen, H., Lenstra, H.W., Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout, : (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068. Springer, Berlin 1984, 33–62 (1983)
Gauss, D.F.: Disquisitiones arithmeticae, (1801)
Chowla, S.: An extension of heilbronn’s class number theorem. Quarterly J. Math. 5, 304–307 (1934)
Boyd, David W.., Kisilevsky, H.: On the exponent of the ideal class groups of complex quadratic fields. Proc. Amer. Math. Soc. 31, 433–436 (1972)
Heath-Brown, D.R.: Imaginary quadratic fields with class group exponent 5. Forum Math. 20(2), 275–283 (2008)
Weinberger, P.J.: Exponents of the class groups of complex quadratic fields. Acta Arith. 22, 117–124 (1973)
Watkins, Mark: Class numbers of imaginary quadratic fields. Math. Comp. 73(246), 907–938 (2004)
Ranum, Arthur: The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group. Trans. Amer. Math. Soc. 8(1), 71–91 (1907)
Holmin, S., Kurlberg, P.: List of \(\cal{F} (g)\) for all noncyclic abelian groups of odd prime power order \(< 10^6\), Available online at: (https://people.kth.se/ kurlberg/class_group_data/noncyclic_class_groups.txt)
Holmin, S., Kurlberg, P.: List of \(\cal{F}(g)\) for all noncyclic abelian groups of odd prime power order \(< 10^6\), Available online at: (https://people.kth.se/~kurlberg/class_group_data/class_group_orders.txt)
Holmin, S., Kurlberg, P.: List of \(\cal{F} (g)\) for all noncyclic abelian groups of odd prime power order \(< 10^6\), Available online at: (https://people.kth.se/~kurlberg/class_group_data/discriminants_of_noncyclic_groups.txt)
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Petersen, K.L., Sellers, J.A. Partitions associated to class groups of imaginary quadratic number fields. Aequat. Math. 97, 63–74 (2023). https://doi.org/10.1007/s00010-022-00899-x
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DOI: https://doi.org/10.1007/s00010-022-00899-x