Taiwanese Journal of Mathematics

ENUMERATION PROBLEMS FOR A LINEAR CONGRUENCE EQUATION

Wun-Seng Chou, Tian-Xiao He, and Peter Shiue

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Abstract

Let $m \geq 2$ and $r \geq 1$ be integers and let $c \in Z_m = \{0, 1,\dots, m - 1\}$. In this paper, we give an upper bound and a lower bound for the number of unordered solutions $x_1, \dots, x_n \in Z_m$ of the congruence $x_1 + x_2 + \cdots + x_r \equiv c \mod m$. Exact formulae are also given when $m$ or $r$ is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has relationship with the restricted integer partition.

Article information

Source
Taiwanese J. Math., Volume 18, Number 1 (2014), 265-275.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706342

Digital Object Identifier
doi:10.11650/tjm.18.2014.2295

Mathematical Reviews number (MathSciNet)
MR3162124

Zentralblatt MATH identifier
1357.05006

Subjects
Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A17: Partitions of integers [See also 11P81, 11P82, 11P83] 05A19: Combinatorial identities, bijective combinatorics 11P81: Elementary theory of partitions [See also 05A17] 11P83: Partitions; congruences and congruential restrictions

Keywords
congruence Catalan number generalized Catalan number integer partition

Citation

Chou, Wun-Seng; He, Tian-Xiao; Shiue, Peter. ENUMERATION PROBLEMS FOR A LINEAR CONGRUENCE EQUATION. Taiwanese J. Math. 18 (2014), no. 1, 265--275. doi:10.11650/tjm.18.2014.2295. https://projecteuclid.org/euclid.twjm/1499706342


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