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Article

The Cranks for 5-Core Partitions

The University of Tennessee at Martin, Martin, TN 38238, USA
Axioms 2012, 1(3), 372-383; https://doi.org/10.3390/axioms1030372
Submission received: 7 August 2012 / Revised: 21 September 2012 / Accepted: 26 November 2012 / Published: 3 December 2012

Abstract

:
It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5n + 4 and 25n + 24 into 5 and 25 classes of equal size, respectively. In this paper we will develop the cranks that can be used to sort the 5-core partitions of 5kn + 5k − 1 into 5k classes of equal size.

1. Introduction

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 n × 1 = 0 n Z t q t 2 n + b × n where the vector 1 = ( 1 , 1 , , 1 ) in Z t and b = ( 0 , 1 , , t 1 ) [3]. In [3] Garvan, Stanton, and Kim showed that the statistic 4n0 + n1 + n3 + 4n4 (mod 5), where the ni’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 5kn + 5k − 1 will be given using techniques similar to those used by Garvan, Stanton, and Kim.

2. Description of the Crank

For ease of working with the vector 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
G ( a , b , c , d ) = 5 a 2 + 5 b 2 + 5 c 2 + 5 d 2 + 5 a b + 5 a c + 5 a d + 5 b c + 5 b d + 5 c d 4 a 3 b 2 c d
Thus the 5-cores of integers of the form 5n + 4 are associated with the values of a, b, c, and d satisfying 4 a 3 b 2 c d 4 ( 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).
H ( A , B , C , D ) = 25 A 2 + 10 B 2 + 50 C 2 + 25 D 2 25 A B 25 B C 25 C D + 15 A 10 B + 15 D + 4
Note that A, B, C, and D are integers since
( A , B , C , D ) = ( ( a , b , c , d ) γ ) T 1 = ( 3 a + b + c + 6 M , 4 a 5 b 3 c 3 d + 3 , 2 a + b + c + 4 M , M )
where T = ( 1 2 0 2 0 1 1 1 1 1 4 1 2 1 1 2 ) , γ = ( 0 , 0 , 0 , 1 ) , and M = 4 a 3 b 2 c d + 1 5 .
Theorem 1.1
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) = ( ( A , B m 5 , C , D ) λ m ) U m where B m   ( mod   5 ) and
λ 0 = ( 0 , 0 , 0 , 0 ) U 0 = ( 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 ) U 0 1 = ( 2 1 1 0 1 0 0 0 3 1 2 1 2 1 1 1 )
λ 1 = ( 0 , 0 , 0 , 0 ) U 1 = ( 1 0 1 0 4 1 1 1 1 0 1 1 0 1 0 0 ) U 1 1 = ( 2 1 1 1 0 0 0 1 3 1 1 1 5 2 3 2 )
λ 2 = ( 2 , 1 , 1 , 1 ) U 2 = ( 0 1 0 1 1 4 1 1 1 1 0 1 1 0 1 0 ) U 2 1 = ( 5 2 3 2 3 1 2 1 5 2 3 1 2 1 2 1 )
λ 3 = ( 1 , 0 , 1 , 0 ) U 3 = ( 0 0 1 0 1 1 4 1 1 1 1 0 0 1 0 1 ) U 3 1 = ( 2 1 2 1 3 1 1 1 1 0 0 0 3 1 1 0 )
λ 4 = ( 2 , 0 , 1 , 0 ) U 4 = ( 0 1 0 0 1 4 1 1 0 1 1 1 1 0 1 0 ) U 4 1 = ( 3 1 1 0 1 0 0 0 3 1 1 1 2 1 2 1 )
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 m   ( 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.
Theorem 1.2
The 5-core partitions of 5kn + 5k − 1 can be sorted into 5k 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 5k solutions of G(a, b, c, d) = 5kn + 5k − 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) = 5kn + 5k − 1. These k-tuples can be used to sort the solutions of G(a, b, c, d) = 5kn + 5k − 1 into 5k classes of equal size.

3. An Illustration of the Crank

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.
Table 1. Solutions corresponding to 5-cores of 14.
Table 1. Solutions corresponding to 5-cores of 14.
Solutions of G(a, b, c, d) = 2Solutions of H(A, B, C, D) = 14Solutions of G(a, b, c, d) = 14Congruence class of B (mod 5)
(0, 1, 0, 0)(−1, 0, 0, 0)(−1, 2, 0, −1)0
(0, 1, 0, −1)(2, 0, 0, −2)1
(1, 2, 1, 0)(0, −1, 2, 0)2
(4, 8, 2, 1)(0, −1, −1, 1)3
(1, 4, 1, 0)(0, 1, 0, −2)4
(1, 0, 0, −1)(0, 0, 0, −1)(2, −1, 1, −1)0
(3, 6, 2, 1)(−1, −1, 1, 1)1
(1, 2, 0, 0)(1, 0, −2, 1)2
(−4, −7, −2, −1)(0, 2, 0, 0)3
(−3, −6, −2, −1)(1, 1, −1, 1)4
Table 2. Solutions corresponding to 5-cores of 74.
Table 2. Solutions corresponding to 5-cores of 74.
Solutions of G(a, b, c, d) = 14Solutions of H(A, B, C, D) = 74Solutions of G(a, b, c, d) = 742-Tuples showing congruence classes of B’s mod 5
(−1, 2, 0, −1)(−6, −10, −2, −1)(−2, 3, 3, −1)(0, 0)
(7, 16, 4, 1)(1, −1, −1, −3)(0, 1)
(−3, −8, −2, −2)(3, −2, 2, 1)(0, 2)
(6, 13, 4, 3)(−4, 0, 0, 2)(0, 3)
(1, 4, 0, −1)(3, 1, −3, −3)(0, 4)
(2, 0, 0, −2)(0, 0, 0, −2)(4, −2, 2, −3)(1, 0)
(6, 11, 4, 2)(−2, −3, 3, 2)(1, 1)
(4, 7, 1, 1)(1, −1, −4, 3)(1, 2)
(−9, −17, −5, −2)(0, 4, −1, 1)(1, 3)
(−8, −16, −5, −2)(1, 3, −2, 2)(1, 4)
(0, −1, 2, 0)(−5, −10, −4, −2)(3, 2, −4, 1)(2, 0)
(−6, −9, −2, −1)(−2, 4, 2, −2)(2, 1)
(5, 12, 3, 0)(2, −1, 0, −4)(2, 2)
(0, −2, 0, −1)(2, −3, 3, 1)(2, 3)
(−3, −6, −1, −2)(2, −1, 4, −2)(2, 4)
(0, −1, −1, 1)(6, 10, 3, 2)(−1, −3, 0, 4)(3, 0)
(−2, −4, −2, 0)(0, 2, −4, 3)(3, 1)
(−8, −13, −4, −2)(0, 5, −1, −2)(3, 2)
(0, 3, 1, −1)(1, 1, 2, −5)(3, 3)
(8, 14, 4, 2)(0, −4, 0, 3)(3, 4)
(0, 1, 0, −2)(−5, −10, −2, −2)(1, 0, 4, −1)(4, 0)
(10, 21, 6, 3)(−2, −2, 0, 0)(4, 1)
(−3, −8, −3, −2)(4, −1, −2, 2)(4, 2)
(−2, −2, 0, 1)(−4, 3, 1, 1)(4, 3)
(−3, −6, −3, −2)(4, 1, −4, 0)(4, 4)
(2, −1, 1, −1)(0, 0, −1, −2)(5, −1, −2, −2)(0, 0)
(−2, −4, 0, 0)(−2, 0, 4, 1)(0, 1)
(8, 17, 4, 2)(0, −1, −3, 0)(0, 2)
(−8, −17, −5, −3)(3, 1, 0, 1)(0, 3)
(−8, −16, −4, −2)(0, 2, 2, 1)(0, 4)
(−1, −1, 1, 1)(−2, −5, −2, 0)(0, 1, −3, 4)(1, 0)
(−6, −9, −3, −1)(−1, 5, −2, −1)(1, 1)
(−3, −3, −1, −2)(2, 2, 1, −5)(1, 2)
(4, 8, 3, 0)(1, −3, 4, −2)(1, 3)
(5, 9, 3, 0)(2, −4, 3, −1)(1, 4)
(1, 0, −2, 1)(10, 20, 6, 3)(−2, −3, 1, 1)(2, 0)
(−1, −4, −2, −1)(3, −1, −3, 3)(2, 1)
(−5, −8, −2, 0)(−3, 4, 0, 1)(2, 2)
(0, 3, 0, -1)(2, 2, −2, −4)(2, 3)
(7, 14, 4, 3)(−3, −1, −1, 3)(2, 4)
(0, 2, 0, 0)(−2, 0, 0, 0)(−2, 4, 0, −3)(3, 0)
(0, 1, 0, −2)(4, −1, 1, −4)(3, 1)
(4, 7, 3, 1)(−1, −3, 4, 1)(3, 2)
(7, 13, 3, 2)(0, −2, −3, 3)(3, 3)
(0, 4, 1, 0)(−1, 3, 0, −4)(3, 4)
(1, 1, −1, 1)(6, 15, 4, 2)(−2, 1, −1, −2)(4, 0)
(−4, −9, −3, −3)(5, −1, 0, −1)(4, 1)
(3, 7, 3, 2)(−4, 0, 3, 1)(4, 2)
(4, 8, 1, 0)(3, −1, −4, 0)(4, 3)
(3, 9, 3, 2)(−4, 2, 1, −1)(4, 4)
Table 3. Solutions corresponding to 5-cores of 74.
Table 3. Solutions corresponding to 5-cores of 74.
Solutions of G(a, b, c, d) = 74Solutions of H(A, B, C, D) = 374Solutions of G(a, b, c, d) = 3743-Tuples showing congruence classes of B’s mod 5
(−2, 3, 3, −1)(−18, −30, −9, −4)(−1, 11, −2, −4)(0, 0, 0)
(0, 6, 2, −2)(2, 2, 4, −11)(0, 0, 1)
(10, 17, 6, 0)(4, −9, 7, −2)(0, 0, 2)
(14, 23, 7, 5)(−3, −7, 0, 9)(0, 0, 3)
(−6, −6, −2, −4)(4, 4, 2, −11)(0, 0, 4)
(1, −1, −1, −3)(0, −5, 0, −2)(4, −7, 7, 2)(0, 1, 0)
(16, 31, 9, 7)(−7, −3, −2, 7)(0, 1, 1)
(−11, −23, −9, −4)(6, 4, −9, 3)(0, 1, 2)
(−14, −22, −5, −2)(−5, 9, 4, −4)(0, 1, 3)
(−3, −11, −5, −2)(6, −2, −7, 7)(0, 1, 4)
(3, −2, 2, 1)(4, 10, 0, −1)(6, 1, −9, −3)(0, 2, 0)
(−17, −34, −8, −5)(1, 3, 7, −1)(0, 2, 1)
(19, 42, 12, 6)(−5, −2, 0, −3)(0, 2, 2)
(−6, −17, −6, −5)(10, −4, −2, 2)(0, 2, 3)
(−9, −16, −2, −1)(−5, 3, 9, −1)(0, 2, 4)
(−4, 0, 0, 2)(−4, −10, −2, 2)(−6, 2, 0, 9)(0, 3, 0)
(−2, 1, −2, 0)(0, 7, −9, −2)(0, 3, 1)
(−18, −33, −9, −7)(5, 5, 4, −7)(0, 3, 2)
(15, 33, 11, 4)(−4, −4, 7, −5)(0, 3, 3)
(18, 34, 9, 2)(5, −9, 0, −2)(0, 3, 4)
(3, 1, −3, −3)(8, 15, 6, 0)(2, −7, 9, −4)(0, 4, 0)
(18, 31, 9, 5)(−1, −9, 0, 7)(0, 4, 1)
(−5, −13, −5, 0)(0, 2, −7, 9)(0, 4, 2)
(−14, −22, −7, −2)(−3, 11, −4, −2)(0, 4, 3)
(−5, −11, −5, 0)(0, 4, −9, 7)(0, 4, 4)
(4, −2, 2, −3)(−2, −5, −3, −5)(11, −3, −2, −5)(1, 0, 0)
(1, 1, 3, 2)(−6, −2, 9, 3)(1, 0, 1)
(16, 32, 7, 4)(1, −3, −8, 2)(1, 0, 2)
(−20, −42, −12, −6)(4, 4, 0, 3)(1, 0, 3)
(−20, −41, −11, −5)(1, 5, 2, 3)(1, 0, 4)
(−2, −3, 3, 2)(−6, −15, −6, −1)(2, 2, −8, 8)(1, 1, 0)
(−15, −24, −7, −2)(−4, 11, −2, −2)(1, 1, 1)
(−2, 2, 0, −3)(4, 3, 1, −11)(1, 1, 2)
(5, 8, 4, −1)(3, −7, 9, −3)(1, 1, 3)
(6, 9, 4, −1)(4, −8, 8, −2)(1, 1, 4)
(1, −1, −4, 3)(21, 40, 12, 7)(−5, −7, 1, 5)(1, 2, 0)
(−5, −14, −6, −2)(5, 0, −8, 7)(1, 2, 1)
(−14, −23, −6, −1)(−6, 10, 0, 0)(1, 2, 2)
(1, 8, 1, −2)(4, 3, −2, −10)(1, 2, 3)
(18, 34, 10, 7)(−6, −5, −1, 7)(1, 2, 4)
(0, 4, −1, 1)(1, 10, 3, 2)(−6, 7, 0, −6)(1, 3, 0)
(−2, −4, −2, −5)(10, −3, 1, −7)(1, 3, 1)
(7, 12, 6, 3)(−5, −5, 9, 3)(1, 3, 2)
(15, 28, 6, 4)(1, −4, −8, 5)(1, 3, 3)
(3, 14, 4, 2)(−5, 6, 0, −7)(1, 3, 4)
(1, 3, −2, 2)(9, 25, 7, 4)(−6, 4, −1, −5)(1, 4, 0)
(−6, −14, −5, −6)(11, −3, 0, −4)(1, 4, 1)
(6, 12, 6, 4)(−8, −2, 8, 3)(1, 4, 2)
(12, 23, 4, 2)(4, −3, −9, 2)(1, 4, 3)
(6, 19, 6, 4)(−8, 5, 1, −4)(1, 4, 4)
(3, 2, −4, 1)(18, 40, 12, 5)(−4, −3, 3, −5)(2, 0, 0)
(1, −4, −2, −3)(9, −7, −1, 3)(2, 0, 1)
(1, 2, 2, 4)(−9, 2, 2, 7)(2, 0, 2)
(0, 3, −2, −1)(4, 4, −10, −2)(2, 0, 3)
(5, 14, 4, 5)(−9, 5, −3, 3)(2, 0, 4)
(−2, 4, 2, −2)(−18, −30, −8, −4)(−2, 10, 2, −5)(2, 1, 0)
(8, 21, 6, 0)(2, −1, 3, −10)(2, 1, 1)
(6, 7, 3, −1)(5, −9, 6, 1)(2, 1, 2)
(13, 23, 7, 6)(−6, −4, −1, 9)(2, 1, 3)
(−6, −6, −3, −4)(5, 5, −2, −10)(2, 1, 4)
(2, −1, 0, −4)(−3, −10, −2, −4)(7, −6, 6, −1)(2, 2, 0)
(16, 31, 10, 7)(−8, −4, 2, 6)(2, 2, 1)
(−3, −8, −5, −2)(6, 1, −10, 4)(2, 2, 2)
(−18, −32, −8, −3)(−4, 9, 3, −1)(2, 2, 3)
(−11, −26, −9, −4)(6, 1, −6, 6)(2, 2, 4)
(2, −3, 3, 1)(0, 0, −3, −2)(7, 1, −10, 0)(2, 3, 0)
(−18, −34, −8, −4)(−2, 6, 6, −1)(2, 3, 1)
(16, 37, 10, 4)(−2, −1, −1, −6)(2, 3, 2)
(−6, −17, −5, −5)(9, −5, 2, 1)(2, 3, 3)
(−8, −16, −2, −2)(−2, 0, 10, −1)(2, 3, 4)
(2, −1, 4, −2)(−11, −20, −8, −6)(9, 4, −6, −5)(2, 4, 0)
(−6, −9, 0, −1)(−4, 2, 10, −4)(2, 4, 1)
(21, 42, 11, 4)(2, −7, −2, −2)(2, 4, 2)
(−8, −22, −6, −3)(4, −3, 1, 7)(2, 4, 3)
(−19, −36, −9, −6)(2, 5, 6, −4)(2, 4, 4)
(−1, −3, 0, 4)(9, 15, 3, 4)(−2, −2, −7, 9)(3, 0, 0)
(−18, −34, −11, −4)(1, 9, −6, 2)(3, 0, 1)
(−8, −8, −2, −2)(−2, 8, 2, −9)(3, 0, 2)
(6, 13, 4, −2)(6, −5, 5, −8)(3, 0, 3)
(16, 29, 10, 4)(−2, −9, 7, 2)(3, 0, 4)
(0, 2, −4, 3)(16, 35, 11, 7)(−9, −1, 2, 1)(3, 1, 0)
(−3, −9, −5, −4)(10, −2, −7, 1)(3, 1, 1)
(−10, −18, −3, 0)(−7, 5, 6, 2)(3, 1, 2)
(12, 28, 6, 2)(2, 0, −6, −5)(3, 1, 3)
(18, 39, 11, 7)(−7, −1, −2, 1)(3, 1, 4)
(0, 5, −1, −2)(−6, −5, 0, −1)(−4, 6, 6, −8)(3, 2, 0)
(13, 26, 7, 0)(6, −7, 2, −6)(3, 2, 1)
(4, 2, 2, 1)(0, −7, 5, 7)(3, 2, 2)
(9, 18, 4, 5)(−5, 1, −7, 7)(3, 2, 3)
(−4, −1, −2, −1)(0, 8, −6, −6)(3, 2, 4)
(1, 1, 2, −5)(−15, −30, −8, −7)(7, 1, 5, −5)(3, 3, 0)
(17, 36, 12, 6)(−7, −4, 6, −1)(3, 3, 1)
(6, 7, 0, −1)(8, −6, −6, 4)(3, 3, 2)
(−11, −22, −5, 0)(−6, 5, 2, 6)(3, 3, 3)
(−18, −36, −12, −7)(8, 5, −5, −1)(3, 3, 4)
(0, −4, 0, 3)(10, 15, 3, 3)(1, −5, −6, 9)(3, 4, 0)
(−15, −29, −9, −2)(−2, 8, −5, 5)(3, 4, 1)
(−8, −8, −3, −2)(−1, 9, −2, −8)(3, 4, 2)
(−2, −2, 0, −4)(6, −2, 6, −9)(3, 4, 3)
(12, 19, 7, 3)(−1, −9, 6, 5)(3, 4, 4)
(1, 0, 4, −1)(−12, −20, −8, −5)(6, 7, −7, −5)(4, 0, 0)
(−9, −14, −2, −3)(−1, 3, 9, −7)(4, 0, 1)
(21, 42, 12, 4)(1, −8, 2, −3)(4, 0, 2)
(0, −7, −2, −1)(4, −6, 0, 8)(4, 0, 3)
(−15, −26, −6, −5)(1, 5, 7, −7)(4, 0, 4)
(−2, −2, 0, 0)(−2, −10, −2, 0)(0, −4, 2, 9)(4, 1, 0)
(4, 11, 2, 4)(−6, 5, −7, 4)(4, 1, 1)
(−18, −33, −11, −7)(7, 7, −4, −5)(4, 1, 2)
(−1, 3, 3, 0)(−4, 2, 9, −7)(4, 1, 3)
(10, 14, 3, 0)(7, −9, −2, 4)(4, 1, 4)
(4, −1, −2, 2)(19, 40, 10, 4)(1, −4, −4, −3)(4, 2, 0)
(−12, −29, −8, −5)(6, −2, 2, 4)(4, 2, 1)
(9, 22, 7, 6)(−10, 3, 0, 2)(4, 2, 2)
(−6, −12, −6, −5)(10, 1, −7, −3)(4, 2, 3)
(1, 4, 3, 4)(−10, 3, 4, 4)(4, 2, 4)
(−4, 3, 1, 1)(−12, −20, −5, 0)(−7, 9, 0, 2)(4, 3, 0)
(0, 6, 0, −2)(4, 4, −4, −9)(4, 3, 1)
(−6, −13, −2, −4)(4, −3, 9, −4)(4, 3, 2)
(22, 43, 13, 7)(−5, −7, 2, 3)(4, 3, 3)
(10, 24, 6, 0)(4, −2, 0, −9)(4, 3, 4)
(4, 1, −4, 0)(19, 40, 12, 4)(−1, −6, 4, −5)(4, 4, 0)
(4, 1, 0, −1)(6, −8, 0, 6)(4, 4, 1)
(1, 2, 1, 4)(−8, 3, −2, 8)(4, 4, 2)
(−8, −12, −6, −3)(4, 7, −9, −3)(4, 4, 3)
(1, 4, 1, 4)(−8, 5, −4, 6)(4, 4, 4)
(5, −1, −2, −2)(13, 25, 7, 0)(6, −8, 3, −5)(0, 0, 0)
(6, 6, 3, 2)(−1, −7, 4, 8)(0, 0, 1)
(6, 12, 2, 4)(−4, 2, −8, 7)(0, 0, 2)
(−20, −37, −12, −6)(4, 9, −5, −2)(0, 0, 3)
(−10, −21, −6, 0)(−4, 5, −3, 8)(0, 0, 4)
(−2, 0, 4, 1)(−14, −25, −9, −3)(1, 9, −8, 1)(0, 1, 0)
(−13, 19, −5, −4)(0, 8, 3, −9)(0, 1, 1)
(10, 22, 7, 0)(3, −5, 6, −8)(0, 1, 2)
(12, 18, 6, 2)(2, −10, 4, 5)(0, 1, 3)
(–2, –1, 1, –3)(3, –1, 8, –9)(0, 1, 4)
(0, −1, −3, 0)(10, 15, 6, 3)(−2, −8, 6, 6)(0, 2, 0)
(9, 16, 3, 4)(−2, −1, −8, 8)(0, 2, 1)
(−20, −38, −12, −5)(2, 9, −5, 1)(0, 2, 2)
(−5, −2, 0, −1)(−3, 7, 3, −9)(0, 2, 3)
(12, 19, 4, 3)(2, −6, −6, 8)(0, 2, 4)
(3, 1, 0, 1)(7, 20, 4, 1)(1, 3, −5, −7)(0, 3, 0)
(−11, −24, −6, −6)(7, −2, 6, −3)(0, 3, 1)
(18, 37, 12, 7)(−8, −4, 4, 2)(0, 3, 2)
(1, −2, −3, −2)(8, −3, −8, 4)(0, 3, 3)
(−6, −6, 0, 1)(−8, 7, 5, −3)(0, 3, 4)
(0, 2, 2, 1)(−6, −5, −3, −1)(−1, 9, −6, −5)(0, 4, 0)
(−11, −19, −5, −6)(6, 2, 5, −9)(0, 4, 1)
(16, 32, 11, 4)(−3, −7, 8, −2)(0, 4, 2)
(12, 18, 4, 2)(4, −8, −4, 7)(0, 4, 3)
(−4, −1, 1, −1)(−3, 5, 6, −9)(0, 4, 4)
(0, 1, −3, 4)(16, 35, 10, 7)(−8, 0, −2, 2)(1, 0, 0)
(−11, −24, −9, −6)(10, 1, −6, 0)(1, 0, 1)
(−6, −8, 0, 1)(−8, 5, 7, −1)(1, 0, 2)
(13, 28, 6, 1)(5, −3, −5, −5)(1, 0, 3)
(18, 39, 12, 7)(−8, −2, 2, 0)(1, 0, 4)
(−1, 5, −2, −1)(−3, 0, 2, 1)(−7, 5, 7, −5)(1, 1, 0)
(13, 26, 6, 0)(7, −6, −2, −5)(1, 1, 1)
(−4, −13, −2, −1)(0, −4, 6, 6)(1, 1, 2)
(13, 28, 7, 6)(−6, 1, −6, 4)(1, 1, 3)
(4, 14, 2, 1)(0, 5, −7, −5)(1, 1, 4)
(2, 2, 1, −5)(−11, −20, −5, −6)(6, 1, 6, −8)(1, 2, 0)
(18, 36, 12, 5)(−4, −7, 7, −1)(1, 2, 1)
(9, 12, 2, 1)(5, −7, −5, 7)(1, 2, 2)
(−11, −22, −6, 0)(−5, 6, −2, 7)(1, 2, 3)
(−19, −36, −12, −6)(5, 8, −6, −1)(1, 2, 4)
(1, −3, 4, −2)(−11, −25, −9, −6)(10, 0, −5, 1)(1, 3, 0)
(−4, −4, 1, 2)(−9, 5, 6, 0)(1, 3, 1)
(10, 22, 4, 0)(6, −2, −6, −5)(1, 3, 2)
(−12, −27, −6, −4)(2, −1, 7, 2)(1, 3, 3)
(−14, −31, −8, −6)(6, −1, 5, 0)(1, 3, 4)
(2, −4, 3, −1)(−3, −10, −5, −4)(10, −3, −6, 2)(1, 4, 0)
(−8, −14, −2, 1)(−8, 5, 5, 3)(1, 4, 1)
(9, 22, 4, 1)(3, 1, −7, −5)(1, 4, 2)
(−15, −32, −8, −6)(5, 0, 6, −1)(1, 4, 3)
(−11, −26, −6, −4)(3, −2, 6, 3)(1, 4, 4)
(−2, −3, 1, 1)(−2, −10, −3, 0)(1, −3, −2, 10)(2, 0, 0)
(−4, −4, −2, 2)(−6, 8, −6, 3)(2, 0, 1)
(−14, −23, −8, −6)(6, 7, −3, −8)(2, 0, 2)
(0, 3, 3, −1)(−1, −1, 10, −7)(2, 0, 3)
(10, 14, 4, 0)(6, −10, 2, 3)(2, 0, 4)
(3, −1, −3, 3)(22, 45, 12, 6)(−2, −5, −3, 0)(2, 1, 0)
(−12, −29, −9, −5)(7, −1, −2, 5)(2, 1, 1)
(1, 7, 3, 4)(−10, 6, 1, 1)(2, 1, 2)
(−2, −2, −3, −4)(9, 1, −6, −6)(2, 1, 3)
(9, 19, 7, 6)(−10, 0, 3, 5)(2, 1, 4)
(−3, 4, 0, 1)(−8, −10, −2, 1)(−8, 9, 1, −1)(2, 2, 0)
(1, 6, 0, −3)(7, 1, −3, −9)(2, 2, 1)
(−3, −8, 0, −2)(1, −4, 10, −1)(2, 2, 2)
(22, 43, 12, 7)(−4, −6, −2, 4)(2, 2, 3)
(9, 24, 6, 1)(1, 1, −1, −9)(2, 2, 4)
(2, 2, −2, −4)(0, 0, 2, −2)(2, −4, 10, −5)(2, 3, 0)
(22, 41, 12, 6)(−2, −9, 1, 4)(2, 3, 1)
(−4, −13, −5, −1)(3, −1, −6, 9)(2, 3, 2)
(−11, −17, −5, 0)(−6, 10, −3, 1)(2, 3, 3)
(−8, −16, −7, −2)(3, 5, −10, 4)(2, 3, 4)
(−3, −1, −1, 3)(4, 5, 2, 4)(−6, −1, −1, 10)(2, 4, 0)
(−6, −9, −5, −1)(1, 7, −10, 1)(2, 4, 1)
(−19, −33, −9, −6)(2, 8, 3, −7)(2, 4, 2)
(12, 28, 9, 2)(−1, −3, 6, −8)(2, 4, 3)
(21, 39, 11, 4)(2, −10, 1, 1)(2, 4, 4)
(−2, 4, 0, −3)(−14, −25, −5, −3)(−3, 5, 8, −3)(3, 0, 0)
(19, 41, 11, 4)(0, −4, −1, −5)(3, 0, 1)
(−6, −18, −5, −4)(7, −5, 2, 4)(3, 0, 2)
(8, 18, 6, 6)(−10, 2, 0, 5)(3, 0, 3)
(−2, −1, −3, −3)(7, 3, −8, −5)(3, 0, 4)
(4, −1, 1, −4)(−2, −5, −2, −5)(10, −4, 2, −6)(3, 1, 0)
(9, 16, 7, 4)(−6, −5, 8, 4)(3, 1, 1)
(12, 22, 4, 3)(2, −3, −9, 5)(3, 1, 2)
(−21, −42, −12, −5)(1, 7, −1, 3)(3, 1, 3)
(−20, −41, −12, −5)(2, 6, −2, 4)(3, 1, 4)
(−1, −3, 4, 1)(−9, −20, −8, −3)(5, 3, −9, 5)(3, 2, 0)
(−15, −24, −6, −2)(−5, 10, 2, −3)(3, 2, 1)
(6, 17, 4, −1)(4, 0, 0, −10)(3, 2, 2)
(1, −2, 1, −2)(4, −7, 8, 0)(3, 2, 3)
(−2, −6, 0, −3)(4, −5, 9, −3)(3, 2, 4)
(0, −2, −3, 3)(17, 30, 9, 6)(−4, −7, 0, 8)(3, 3, 0)
(−6, −14, −6, −1)(2, 3, −9, 7)(3, 3, 1)
(−17, −28, −8, −3)(−3, 11, −1, −3)(3, 3, 2)
(1, 8, 2, −2)(3, 2, 2, −11)(3, 3, 3)
(19, 34, 10, 6)(−3, −8, 0, 7)(3, 3, 4)
(−1, 3, 0, −4)(−13, −25, −5, −4)(0, 2, 9, −3)(3, 4, 0)
(22, 46, 13, 6)(−3, −5, 0, −2)(3, 4, 1)
(−6, −18, −6, −4)(8, −4, −2, 5)(3, 4, 2)
(0, 3, 2, 4)(−10, 5, 1, 4)(3, 4, 3)
(−6, −11, −6, −4)(8, 3, −9, −2)(3, 4, 4)
(−2, 1, −1, −2)(−6, −15, −2, −1)(−2, −2, 8, 4)(4, 0, 0)
(17, 36, 9, 6)(−4, −1, −6, 2)(4, 0, 1)
(−18, −38, −12, −7)(8, 3, −3, 1)(4, 0, 2)
(1, 8, 4, 3)(−9, 5, 5, −3)(4, 0, 3)
(6, 9, 0, −1)(8, −4, −8, 2)(4, 0, 4)
(5, −1, 0, −1)(9, 20, 4, −1)(7, −3, −3, −7)(4, 1, 0)
(−5, −14, −2, −2)(1, −4, 8, 3)(4, 1, 1)
(18, 37, 10, 7)(−6, −2, −4, 4)(4, 1, 2)
(−15, −32, −11, −6)(8, 3, −6, 2)(4, 1, 3)
(−14, −26, −6, −1)(−6, 7, 3, 3)(4, 1, 4)
(−4, 0, 3, 1)(−15, −30, −9, −2)(−2, 7, −4, 6)(4, 2, 0)
(−6, −4, −2, −1)(−2, 9, −3, −7)(4, 2, 1)
(−5, −8, −2, −5)(7, −1, 5, −9)(4, 2, 2)
(15, 28, 10, 4)(−3, −8, 8, 1)(4, 2, 3)
(7, 14, 4, −2)(7, −6, 4, −7)(4, 2, 4)
(3, −1, −4, 0)(19, 35, 11, 4)(0, −10, 5, 1)(4, 3, 0)
(6, 6, 1, 2)(1, −5, −4, 10)(4, 3, 1)
(−10, −18, −6, 0)(−4, 8, −6, 5)(4, 3, 2)
(−12, −17, −6, −4)(2, 9, −3, −8)(4, 3, 3)
(6, 9, 2, 4)(−4, −1, −5, 10)(4, 3, 4)
(−4, 2, 1, −1)(−15, −30, −7, −2)(−4, 5, 4, 4)(4, 4, 0)
(10, 26, 6, 3)(−2, 3, −5, −5)(4, 4, 1)
(−13, −28, −8, −7)(9, −1, 3, −3)(4, 4, 2)
(13, 28, 10, 6)(−9, −2, 6, 1)(4, 4, 3)
(7, 14, 2, −2)(9, −4, −4, −5)(4, 4, 4)

4. Conclusion

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.

Acknowledgments

I would like to thank the reviewers for their helpful suggestions.

References

  1. Garvan, F.G. More cranks and t-cores. Bull. Aust. Math. Soc. 2001, 63, 379–391. [Google Scholar] [CrossRef]
  2. James, G.; Kerber, A. The Representation Theory of the Symmetric Group; Addison-Wesley: Reading, MA, USA, 1981; pp. 379–391. [Google Scholar]
  3. Garvan, F.G.; Kim, D.; Stanton, D. Cranks and t-cores. Invent. Math. 1990, 101, 1–17. [Google Scholar] [CrossRef]

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Kolitsch, L. The Cranks for 5-Core Partitions. Axioms 2012, 1, 372-383. https://doi.org/10.3390/axioms1030372

AMA Style

Kolitsch L. The Cranks for 5-Core Partitions. Axioms. 2012; 1(3):372-383. https://doi.org/10.3390/axioms1030372

Chicago/Turabian Style

Kolitsch, Louis. 2012. "The Cranks for 5-Core Partitions" Axioms 1, no. 3: 372-383. https://doi.org/10.3390/axioms1030372

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