The Cranks for 5-Core Partitions

A t-core partition of n is a partition of n that contains no hook numbers that are multiples of t [2, 2.7.40]. The generating function for t-core partitions is given by ${\displaystyle \sum _{\begin{array}{l}\overrightarrow{n}\times \overrightarrow{1}=0\\ \overrightarrow{n}\in Z^t\end{array}}{q}^{\frac{t}{2}\Vert \overrightarrow{n}\Vert +\overrightarrow{b}\times \overrightarrow{n}}}$ where the vector $\overrightarrow{1}=(1,1,\dots ,1)$ in $Z^t$ and $\overrightarrow{b}=(0,1,\dots ,t-1)$ [3]. In [3] Garvan, Stanton, and Kim showed that the statistic 4n₀ + n₁ + n₃ + 4n₄ (mod 5), where the n_i’s are the components of the vector in the generating function for 5-cores, can be used to sort the 5-cores of 5n + 4 into 5 classes of equal size. In a sequel to this paper [1] Garvan explicitly describes a crank for the 5-cores of 25n + 24. In this paper a crank for the 5-cores of 5^kn + 5^k − 1 will be given using techniques similar to those used by Garvan, Stanton, and Kim.

For ease of working with the vector $\overrightarrow{n}$ we will write it as (a, b, c, d, e). Using the fact that a + b + c + d + e = 0, the exponent on q in the generating function for the 5-core partitions can be expressed as Thus the 5-cores of integers of the form 5n + 4 are associated with the values of a, b, c, and d satisfying $-4a-3b-2c-d\equiv 4(\mathrm{mod}5)$. Evaluating G(a, b, c, d) with a = A − C − 2D, b = −2A + B − C + D, c = −B + 4C − D, d = 2A − B − C + 2D + 1, we get an expression in A, B, C, and D which we will label as H(A, B, C, D). Note that A, B, C, and D are integers since where $T=\left(\begin{array}{cccc}1& -2& 0& 2\\ 0& 1& -1& -1\\ -1& -1& 4& -1\\ -2& 1& -1& 2\end{array}\right)$, $\mathsf{\gamma }=(0,0,0,1)$, and $M=\frac{-4a-3b-2c-d+1}{5}$.

The 5-core partitions of 5n + 4 corresponding to the vectors (A, B, C, D) can be sorted into 5 classes of equal size by looking at the values of B modulo 5.

To see this, let (A, B, C, D) = $\left(\left(A,\frac{B-m}{5},C,D\right)-{\mathsf{\lambda }}_m\right)U_m$ where $B\equiv m(\mathrm{mod}5)$ and Note that A, B, C, and D are integers and for each of these changes of variable H(A, B, C, D) becomes 5G(A, B, C, D) + 4. Hence G(A, B, C, D) = n and for each solution of this equation we have 5 solutions (A, B, C, D) of H(A, B, C, D) = 5n + 4, one with $B\equiv m(\mathrm{mod}5)$ for each choice of m = 0, 1, 2, 3, 4, which can be transformed to a solution (a, b, c, d) of G(a, b, c, d) = 5n + 4. This completes the proof of the theorem.

The 5-core partitions of 5^kn + 5^k − 1 can be sorted into 5^k classes of equal size.

From the proof of Theorem 1.1 we can transform a solution of G(a, b, c, d) = n into 5 solutions of G(a, b, c, d) = 5n + 4. Each solution of G(a, b, c, d) = 5n + 4 can be transformed into 5 solutions of G(a, b, c, d) = 25n + 24. Iterating this process k times we easily see that a solution of G(a, b, c, d) = n can be transformed into 5^k solutions of G(a, b, c, d) = 5^kn + 5^k − 1. At each stage in the transformation process we can keep track of the congruence class modulo 5 of B to get a k-tuple of values m (mod 5) associated with each solution of G(a, b, c, d) = 5^kn + 5^k − 1. These k-tuples can be used to sort the solutions of G(a, b, c, d) = 5^kn + 5^k − 1 into 5^k classes of equal size.

The following series of Table 1, Table 2 and Table 3 show the 2 solutions of G(a, b, c, d) = 2 transformed into 250 solutions of G(a, b, c, d) = 374. The intermediate solutions of H(A, B, C, D) are shown in order to easily see the classes of B (mod 5) which can be used to sort these 250 solutions into 125 classes of equal size.

Though this crank is not explicit like the ones presented by Garvan, Stanton, and Kim, its iterative nature makes it easy to program using a computer algebra system. I used a simple routine in MAPLE to generate the information included in the tables in the previous section.