Rad HAZU, Matematičke znanosti, Vol. 28 (2024), 49-56.

NEW PARTITION IDENTITIES FOR ODD W ODD

Mirko Primc

Abstract.   In this note we conjecture Rogers-Ramanujan type colored partition identities for an array N_w^odd with odd number of rows w such that the first and the last row consist of even positive integers. In a strange way this is different from the partition identities for the array N_w with odd number of rows w such that the first and the last row consist of odd positive integers - the partition identities conjectured by S. Capparelli, A. Meurman, A. Primc and the author and related to standard representations of the affine Lie algebra of type C_l⁽¹⁾ for w = 2l + 1. The conjecture is based on numerical evidence.

2020 Mathematics Subject Classification.   05A19, 17B67.

Key words and phrases.   Colored partitions, Rogers-Ramanujan type identities.

DOI: https://doi.org/10.21857/9e31lhzl8m

References:

1. G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, 1976.
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2. S. Capparelli, A. Meurman, A. Primc and M. Primc, New partition identities from C_l⁽¹⁾-modules, Glas. Mat. Ser. III 57 (2022), 161-184.
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3. V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed, Cambridge Univ. Press, Cambridge, 1990.
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4. J. Lepowsky and R. L. Wilson, The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), 199-290; II: The case A₁⁽¹⁾, principal gradation, Invent. Math. 79 (1985), 417-442.
   MathSciNet     CrossRef     MathSciNet     CrossRef