On the k-measure of partitions and distinct partitions
Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1353-1361.

The k-measure of an integer partition was recently introduced by Andrews, Bhattacharjee and Dastidar. In this paper, we establish trivariate generating function identities counting both the length and the k-measure for partitions and distinct partitions, respectively. The 2-measure case for partitions extends a result of Andrews, Bhattacharjee and Dastidar.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.253
Classification: 11P84,  05A17
Keywords: partition, distinct partition, generating function, k-measure, Durfee square
Andrews, George E. 1; Chern, Shane 2; Li, Zhitai 1

1 Penn State University Department of Mathematics University Park PA 16802 (USA)
2 Dalhousie University Department of Mathematics and Statistics Halifax Nova Scotia, B3H 4R2 (Canada)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2022__5_6_1353_0,
     author = {Andrews, George E. and Chern, Shane and Li, Zhitai},
     title = {On the $k$-measure of partitions and distinct partitions},
     journal = {Algebraic Combinatorics},
     pages = {1353--1361},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {6},
     year = {2022},
     doi = {10.5802/alco.253},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.253/}
}
TY  - JOUR
AU  - Andrews, George E.
AU  - Chern, Shane
AU  - Li, Zhitai
TI  - On the $k$-measure of partitions and distinct partitions
JO  - Algebraic Combinatorics
PY  - 2022
DA  - 2022///
SP  - 1353
EP  - 1361
VL  - 5
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.253/
UR  - https://doi.org/10.5802/alco.253
DO  - 10.5802/alco.253
LA  - en
ID  - ALCO_2022__5_6_1353_0
ER  - 
%0 Journal Article
%A Andrews, George E.
%A Chern, Shane
%A Li, Zhitai
%T On the $k$-measure of partitions and distinct partitions
%J Algebraic Combinatorics
%D 2022
%P 1353-1361
%V 5
%N 6
%I The Combinatorics Consortium
%U https://doi.org/10.5802/alco.253
%R 10.5802/alco.253
%G en
%F ALCO_2022__5_6_1353_0
Andrews, George E.; Chern, Shane; Li, Zhitai. On the $k$-measure of partitions and distinct partitions. Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1353-1361. doi : 10.5802/alco.253. https://alco.centre-mersenne.org/articles/10.5802/alco.253/

[1] Andrews, George E. q-identities of Auluck, Carlitz, and Rogers, Duke Math. J., Volume 33 (1966), pp. 575-581 | MR | Zbl

[2] Andrews, George E. The Theory of Partitions, Cambridge University Press, Cambridge, 1998

[3] Andrews, George E.; Berndt, Bruce C. Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009

[4] Andrews, George E.; Bhattacharjee, Sreerupa; Dastidar, Manosij G. Sequences in partitions, Integers, Volume 22 (2022), p. Paper No. A32, 9 pp. | MR | Zbl

[5] Bhatnagar, Gaurav A bibasic Heine transformation formula and Ramanujan’s 2 φ 1 transformations, Analytic number theory, modular forms and q-hypergeometric series, Springer, Cham, 2017, pp. 99-122 | DOI | MR | Zbl

[6] Binner, Damanvir S. On k-measures and Durfee squares of partitions (2022) | arXiv

[7] Gasper, George; Rahman, Mizan Basic Hypergeometric Series, Second Edition, Cambridge University Press, Cambridge, 2004 | DOI

[8] Hirschhorn, Michael D. Developments in the Theory of Partitions, Ph. D. Thesis, University of New South Wales (1979) | MR

[9] MacMahon, Percy A. Combinatory Analysis, Vol. 2, A.M.S. Chelsea Publishing, Providence, 1984

[10] Sylvester, James J. A constructive theory of partitions, arranged in three acts, an interact and an exodion, Amer. J. Math., Volume 5 (1882), pp. 251-330 | DOI

Cited by Sources: