Abstract
We consider \(\Delta _2(n)\), the number of broken 2-diamond partitions of n, and give simple proofs of two congruences given by Song Heng Chan.
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1 Introduction
Andrews and Paule [1] introduced the concept of broken k-diamond partitions and showed that the generating function for \(\Delta _k(n)\), the number of broken k-diamond partitions of n, is
The following congruences were proved by Chan [2] and again by Radu [4]:
and
Indeed, Chan generalised these to
and
The object of this note is to give as simple a proof as I can of (1.2)–(1.5).
2 Proofs
We start by noting that the 5–dissection of \(\displaystyle \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}\) is
where
Note that by [3, (34.1.21)]
We have, modulo 5,
Alternatively,
or, again, by [3, (34.1.23)],
Thus, we have
It follows that
and
It is now an easy induction (replace n by \(5n+3\)) to deduce that for \(\alpha \ge 1\),
Since there are no terms on the right in which the power of q is congruent to 2 or 4 modulo 5, we have that for \(\alpha \ge 1\),
and
as claimed.
References
Andrews, G.E., Paule, P.: MacMahon’s partition analysis XI, broken diamonds and modular forms. Acta Arith. 126(3), 281–294 (2007)
Chan, S.H.: Some congruences for Andrews–Paule’s broken 2-diamond partitions. Discret. Math. 308(23), 5735–5741 (2008)
Hirschhorn, M.D.: The power of \(q\). Springer (to appear)
Radu, C.-S.: An algorithmic approach to Ramanujan–Kolberg identities. J. Symb. Comput. 68(1), 225–253 (2015)
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Hirschhorn, M.D. Broken 2-diamond partitions modulo 5. Ramanujan J 45, 517–520 (2018). https://doi.org/10.1007/s11139-016-9863-4
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DOI: https://doi.org/10.1007/s11139-016-9863-4