Abstract: The paper solves the enumeration of the set PP(n) of partitions of in which the nondecreasing sequence of parts p(1), p(2), ... , p(d) is contained in a degree-2 polynomial This is a generalization of the partitions of a number into arithmetic progressions. We also study the problem of dividing n into parts whose differences between consecutive parts are consecutive integers. In particular, we focus on the problem of the sum of consecutive triangular numbers.
|
Keywords and phrases: partition, parabola, arithmetic generated by a sequence.
Received: May 6, 2023; Accepted: July 1, 2023; Published: July 13, 2023
How to cite this article: F. Javier de Vega, On the parabolic partitions of a number, JP Journal of Algebra, Number Theory and Applications 61(2) (2023), 135-169. http://dx.doi.org/10.17654/0972555523015
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] M. McMullen, Playing with blocks, Math Horiz. 25(4) (2018), 14-15. [2] A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Seq. 11(5) (2008), #08.5.4. [3] A. O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10 (2010), 73-82. [4] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2022. https://oeis.org. [5] D. Subramaniam, E. Treviño and P. Pollack, On sums of consecutive triangular numbers, Proceedings of the Integers Conference 2018, Integers 20A (2020), #A15. [6] F. Javier de Vega, An extension of Furstenberg’s theorem of the infinitude of primes, JP Journal of Algebra Number Theory and Applications 53(1) (2022), 21-43. [7] F. Javier de Vega, A complete solution of the partition of a number into arithmetic progressions, JP Journal of Algebra Number Theory and Applications 53(2) (2022), 109-122.
|