Parity Indices and Two-Line Matrix Representation for Partitions

CRAVEIRO, I.M.; WAGNER, A.; DOMINGUES, D.

doi:10.5540/tema.2015.016.03.0253

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Articles • TEMA (São Carlos) 16 (3) • Dec 2015 • https://doi.org/10.5540/tema.2015.016.03.0253 copy

Parity Indices and Two-Line Matrix Representation for Partitions

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In this work we present a solution for Andrews's Problem 5¹1 G.E. Andrews. Parity in partition identities. Ramanujan Journal, 23 (2010), 45-90. by establishing a bijection between the sets and defined in Fine's Theorem⁸8 I. Pak. On Fine's partition theorems, Dyson, Andrews and missed opportunities. Mathematical Intel- ligencer, 25(1) (2003), p. 10. and the sets of partitions indexed by their lower parity index¹1 G.E. Andrews. Parity in partition identities. Ramanujan Journal, 23 (2010), 45-90.. We also solve Andrews's Problem 6, conjectured in¹1 G.E. Andrews. Parity in partition identities. Ramanujan Journal, 23 (2010), 45-90..

Keywords:
partition; parity index; mock theta function

11	12
10 + 1	11 + 1
9 + 2	10 + 2
8 + 3	9 + 3
7 + 4	8 + 4
6 + 5	7 + 5
8 + 2 + 1	9 + 2 + 1
7 + 3 + 1	8 + 3 + 1
6 + 4 + 1	7 + 4 + 1
6 + 3 + 2	7 + 3 + 2
5 + 4 + 2	6 + 5 + 1
5 + 3 + 2 + 1	6 + 4 + 2
	5 + 4 + 3
	6 + 3 + 2 + 1
	5 + 4 + 2 + 1

m	r	po(r, m, 11)	po(r, m, 11)(-1)r	m	r	po(r, m, 12)	po(r, m, 12)(-1)r
1	0	0	0	1	0	1	1
1	1	1	-1	1	1	0	0
2	0	0	0	2	0	2	2
2	1	2	-2	2	1	3	-3
2	2	3	3	2	2	0	0
3	0	0	0	3	0	1	1
3	1	2	-2	3	1	1	-1
3	2	3	3	3	2	2	2
3	3	0	0	3	3	3	-3
4	0	0	0	4	0	0	0
4	1	0	0	4	1	0	0
4	2	0	0	4	2	0	0
4	3	1	-1	4	3	1	-1
4	4	0	0	4	4	1	1

partitions λ of 7	Ilo(λ) = r	m	w(λ) = (-1)m + r
7	1	1	(-1)1 + 1 = 1
6 + 1	2	2	(-1)2 + 2 = 1
5 + 2	1	2	(-1)1 + 2 = -1
5 + 1 + 1	1	3	(-1)1 + 3 = 1
4 + 3	2	2	(-1)2 + 2 = 1
4 + 2 + 1	2	3	(-1)3 + 2 = -1
4 + 1 + 1 + 1	2	4	(-1)4 + 2 = 1
3 + 3 + 1	1	3	(-1)3 + 1 = 1
3 + 2 + 2	1	3	(-1)3 + 1 = 1
3 + 2 + 1 + 1	3	4	(-1)4 + 3 = -1
3 + 1 + 1 + 1 + 1	1	5	(-1)5 + 1 = 1
2 + 2 + 2 + 1	2	4	(-1)4 + 2 = 1
2 + 2 + 1 + 1 + 1	2	5	(-1)5 + 2 = -1
2 + 1 + 1 + 1 + 1 + 1	2	6	(-1)6 + 2 = 1
1 + 1 + 1 + 1 + 1 + 1 + 1	1	7	(-1)7 + 1 = 1

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[1] Irene Magalhães Craveiro

m	r	po(r, m, 11)	po(r, m, 11)(-1)r	m	r	po(r, m, 12)	po(r, m, 12)(-1)r
1	0	0	0	1	0	1	1
1	1	1	-1	1	1	0	0
2	0	0	0	2	0	2	2
2	1	2	-2	2	1	3	-3
2	2	3	3	2	2	0	0
3	0	0	0	3	0	1	1
3	1	2	-2	3	1	1	-1
3	2	3	3	3	2	2	2
3	3	0	0	3	3	3	-3
4	0	0	0	4	0	0	0
4	1	0	0	4	1	0	0
4	2	0	0	4	2	0	0
4	3	1	-1	4	3	1	-1
4	4	0	0	4	4	1	1

m	r	po(r, m, 11)	po(r, m, 11)(-1)r	m	r	po(r, m, 12)	po(r, m, 12)(-1)r
1	0	0	0	1	0	1	1
1	1	1	-1	1	1	0	0
2	0	0	0	2	0	2	2
2	1	2	-2	2	1	3	-3
2	2	3	3	2	2	0	0
3	0	0	0	3	0	1	1
3	1	2	-2	3	1	1	-1
3	2	3	3	3	2	2	2
3	3	0	0	3	3	3	-3
4	0	0	0	4	0	0	0
4	1	0	0	4	1	0	0
4	2	0	0	4	2	0	0
4	3	1	-1	4	3	1	-1
4	4	0	0	4	4	1	1

m	r	po(r, m, 11)	po(r, m, 11)(-1)r	m	r	po(r, m, 12)	po(r, m, 12)(-1)r
1	0	0	0	1	0	1	1
1	1	1	-1	1	1	0	0
2	0	0	0	2	0	2	2
2	1	2	-2	2	1	3	-3
2	2	3	3	2	2	0	0
3	0	0	0	3	0	1	1
3	1	2	-2	3	1	1	-1
3	2	3	3	3	2	2	2
3	3	0	0	3	3	3	-3
4	0	0	0	4	0	0	0
4	1	0	0	4	1	0	0
4	2	0	0	4	2	0	0
4	3	1	-1	4	3	1	-1
4	4	0	0	4	4	1	1