On a generalized crank for k-colored partitions

被引：25

作者：

Fu, Shishuo ^{[1
]}

Tang, Dazhao ^{[2
]}

机构：

[1] Chongqing Univ, Coll Math & Stat, Huxi Campus LD506, Chongqing 401331, Peoples R China

[2] Chongqing Univ, Coll Math & Stat, Huxi Campus LD208, Chongqing 401331, Peoples R China

来源：

JOURNAL OF NUMBER THEORY | 2018年 / 184卷

基金：

中国国家自然科学基金;

关键词：

Colored partitions; Partition statistics; Generalized crank; Generalized crank moments; Arithmetic properties; RANKS; CONJECTURE; SYMMETRIES; DYSONS;

D O I：

10.1016/j.jnt.2017.08.033

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

A generalized crank (k-crank) for k-colored partitions is introduced. Following the work of Andrews Lewis and Ji-Zhao, we derive two results for this newly defined k-crank. Namely, we first obtain some inequalities between the k-crank counts M-k(r,m, n) for m = 2, 3 and 4, then we prove the positivity of symmetrized even k-crank moments weighted by the parity for k = 2 and 3. We conclude with several, remarks on furthering the study initiated here. (C) 2017 Elsevier Inc. All rights reserved.

引用

页码：485 / 497

页数：13

共 24 条

[1]

Andrews G.E., 1975, Theory and application of special functions, V35, P191

[2] The ranks and cranks of partitions moduli 2, 3, and 4 [J].

Andrews, GA ;

Lewis, R .

JOURNAL OF NUMBER THEORY, 2000, 85 (01) :74-84

[3] On a conjecture of Peter Borwein [J].

Andrews, GE .

JOURNAL OF SYMBOLIC COMPUTATION, 1995, 20 (5-6) :487-501

[4] DYSONS CRANK OF A PARTITION [J].

ANDREWS, GE ;

GARVAN, FG .

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 1988, 18 (02) :167-171

[5]

Andrews GE., 1976, The Theory of Partitions, Encyclopedia of Mathematics and its Applications

[6] Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks [J].

Andrews, George E. .

INVENTIONES MATHEMATICAE, 2007, 169 (01) :37-73

[7] Relations between the ranks and cranks of partitions [J].

Atkin, AOL ;

Garvan, FG .

RAMANUJAN JOURNAL, 2003, 7 (1-3) :343-366

[8] Some observations on Dyson's new symmetries of partitions [J].

Berkovich, A ;

Garvan, FG .

JOURNAL OF COMBINATORIAL THEORY SERIES A, 2002, 100 (01) :61-93

[9]

Berndt B.C., 1985, Ramanujan's Notebooks

[10]

Bringmann K., 2013, ARXIV13047208

← 1 2 3 →